<PAGE>   1
                                  FORM 10Q
                     SECURITIES AND EXCHANGE COMMISSION
                           Washington, D.C. 20549

            [X] QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d)
                   OF THE SECURITIES EXCHANGE ACT OF 1934
              For the quarterly period ended December 29, 1995
                                             -----------------

                                     OR

           [ ] TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d)
                   OF THE SECURITIES EXCHANGE ACT OF 1934
               For the transition period from       to      
                                              ------   -----

                       Commission File Number 1-6549
                                              ------

                   American Science and Engineering, Inc.          
        ------------------------------------------------------------
           (Exact name of Registrant as specified in its charter)

        Massachusetts                                            04-2240991    
-------------------------------                             -------------------
(State or other jurisdiction of                             (I.R.S. Employer
incorporation or organization)                              Identification No.)

   829 Middlesex Turnpike
   Billerica, Massachusetts                                    01821    
-----------------------------                               ------------
(Address of principal executive offices)                     (Zip Code)

                               (508) 262-8700
            ---------------------------------------------------
            (Registrant's telephone number, including area code)

--------------------------------------------------------------------------------
(Former name, former address and former fiscal year, if changed since last 
                                   report)

Indicate by check mark whether the Registrant (1) has filed all reports 
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of 
1934 during the preceding 12 months, and (2) has been subject to such filing 
requirements for the past 90 days.

                        Yes   X            No      
                            -----             -----

Indicate the number of shares outstanding of each of the issuer's classes of 
common stock, as of the latest practicable date

                                                  Outstanding at
          Class of Common Stock                   December 29, 1995
          ---------------------                   -----------------
          $.66 2/3 par value                      4,416,048


                             Page 1 of 11 Pages
                  The Exhibit Index is Located at Page 10

<PAGE>   2

                 
<TABLE>

                                              AMERICAN SCIENCE AND ENGINEERING, INC.
                                                  PART I - FINANCIAL INFORMATION
                                         CONDENSED CONSOLIDATED STATEMENTS OF OPERATIONS
                                                           (Unaudited)

<CAPTION>
                                                        For The Three Months Ended            For The Nine Months Ended
Dollars and shares in thousands, except               ------------------------------        ------------------------------ 
per share amounts                                     Dec. 29, 1995    Dec. 30, 1994        Dec. 29, 1995    Dec. 30, 1994
                                                      -------------    -------------        -------------    -------------
<S>                                                      <C>               <C>                 <C>              <C>
NET SALES AND CONTRACT REVENUE                           $4,717            $4,020              $12,116          $ 9,941
                                                         ------            ------              -------          -------

COSTS AND EXPENSES:                                             
  Cost of sales and contracts                             3,297             2,685                8,327            6,824
  Selling, general and administrative expenses            1,030             1,112                2,951            2,894
  Research and development                                  113               181                  296              637
                                                         ------            ------              -------          -------
  Total costs and expenses                                4,440             3,978               11,574           10,355
                                                         ------            ------              -------          -------

OPERATING INCOME (LOSS)                                     277                42                  542             (414)
                                                         ------            ------              -------          -------

OTHER INCOME (EXPENSE):                                          
  Interest, net                                              10                46                   48               60
  Other, net                                                (35)              (27)                (131)             (47)
                                                         ------            ------              -------          -------
  Total other income (expense)                              (25)               19                  (83)              13
                                                         ------            ------              -------          -------

INCOME (LOSS) BEFORE PROVISION
   FOR INCOME TAXES                                         252                61                  459             (401)

PROVISION FOR INCOME TAXES                                    -                 -                    -                -
                                                         ------            ------              -------          -------
NET INCOME (LOSS)                                        $  252            $   61               $  459          $  (401)
                                                         ======            ======               ======          =======

Income (loss) per common share and
  common equivalent share                                $  .06            $  .01               $  .10          $  (.10)
                                                         ======            ======               ======          =======

Income (loss) per common share,
  assuming full dilution                                 $  .05            $  .01               $  .10          $  (.10)
                                                         ======            ======               ======          =======

Dividends paid per share                                   NONE              NONE                 NONE             NONE
                                                         ======            ======               ======          =======

Shares used in income (loss) per share and
  common equivalent shares                                4,563             4,248                4,473            4,129
                                                         ======            ======               ======          =======

Shares used in income (loss) per common
  share, assuming full dilution                           4,605             4,248                4,494            4,129
                                                         ======            ======               ======          =======

</TABLE>

              The accompanying notes are an integral part of these condensed 
              consolidated financial statements.

                                     -2-


<PAGE>   3

          
<TABLE>
                                              AMERICAN SCIENCE AND ENGINEERING, INC.
                                              CONDENSED CONSOLIDATED BALANCE SHEETS
<CAPTION>
Dollars in thousands
                                                                         Dec. 29, 1995      Mar. 31, 1995
                                                                         -------------      -------------
                                                                           (Unaudited)
<S>                               <C>                                        <C>               <C>
ASSETS                            CURRENT ASSETS:                       
                                  Cash and cash equivalents                  $   713           $   869
                                  Accounts receivable, net                     5,053             2,831
                                  Unbilled costs and fees, net                 1,242             1,983
                                  Inventories                                  3,610             3,709
                                  Prepaid expenses and other current             
                                    assets                                       208                53
                                  Deferred income taxes                           28                28
                                                                             -------           -------
                                  Total current assets                        10,854             9,473
                                                                             -------           -------

                                  NON CURRENT ASSETS:
                                  Deposits                                       257               207
                                  Property and equipment, net
                                    of accumulated depreciation of
                                    $8,483 at Dec. 29,1995 and 
                                    $8,336 at Mar. 31,1995                       873             1,054
                                                                             -------           -------
                                                                             $11,984           $10,734
                                                                             =======           =======


</TABLE>
       The accompanying notes are an integral part of these condensed 
       consolidated financial statements.


                                     -3-

<PAGE>   4

           
<TABLE>

                                              AMERICAN SCIENCE AND ENGINEERING, INC.
                                        CONDENSED CONSOLIDATED BALANCE SHEETS (CONTINUED)
<CAPTION>
Dollars in thousands
                                                             Dec. 29, 1995             Mar. 31, 1995
                                                             -------------             -------------
                                                              (Unaudited)
<S>                 <C>                                         <C>                       <C>
LIABILITIES &       CURRENT LIABILITIES:
STOCKHOLDERS'       Current maturities of obligations
INVESTMENT           under capital leases                       $    15                   $   160
                    Accounts payable                              1,853                     2,169
                    Accrued legal expenses                          254                       350
                    Accrued salaries and benefits                   594                       557
                    Accrued warranty costs                          209                       152
                    Accrued commissions                              52                       184
                    Accrued rent                                    221                       570
                    Deferred revenue                              1,078                       194
                    Customer deposits                                68                        48
                    Other current liabilities                       366                       473
                                                                -------                   -------
                    TOTAL CURRENT LIABILITIES                     4,710                     4,857
                                                                -------                   -------

                    NONCURRENT LIABILITIES:
                    Obligations under capital leases, net
                      of current maturities                          62                        73
                    Deferred compensation                           191                       184
                    Deferred income taxes                            28                        28
                                                                -------                   -------
                    TOTAL NONCURRENT LIABILITIES                    281                       285
                                                                -------                   -------

                    STOCKHOLDERS' INVESTMENT:                           
                    Preferred stock, no par value
                    Authorized - 100,000 shares
                    Issued - None
                    Common stock, $.66-2/3 par value
                    Authorized - 8,000,000 shares
                    Issued 4,472,926 shares at Dec. 29, 1995
                      and 4,216,475 shares at Mar. 31, 1995       2,985                     2,838
                    Capital in excess of par value               14,406                    13,612
                    Accumulated deficit                          (9,632)                  (10,092)
                                                                -------                   -------
                                                                  7,759                     6,358
                    Note receivable-Officer                        (640)                     (640)
                    Less: treasury stock -
                    67,377 shares at Dec. 29, 1995
                      and Mar. 31, 1995 at cost                    (126)                     (126)
                                                                -------                   -------

                    TOTAL STOCKHOLDERS' INVESTMENT                6,993                     5,592
                                                                -------                   -------
                                                                $11,984                   $10,734
                                                                =======                   =======


</TABLE>
         The accompanying notes are an integral part of these condensed 
         consolidated financial statements.

                                     -4-

<PAGE>   5

<TABLE>

                                                  AMERICAN SCIENCE AND ENGINEERING, INC.
                                             CONDENSED CONSOLIDATED STATEMENTS OF CASH FLOWS
                                                               (Unaudited)
<CAPTION>
Dollars in thousands
                                                                                       For the Nine Months Ended
                                                                                 ------------------------------------
                                                                                 Dec. 29, 1995          Dec. 30, 1994
                                                                                 -------------          -------------
<S>                                                                                 <C>                   <C>
CASH FLOWS FROM OPERATING ACTIVITIES:
Net income (loss)                                                                   $   459               $  (401)
Adjustments to reconcile net income (loss) to net cash used for
   operating activities:
   Depreciation and amortization                                                        148                   198
   Amortization of deferred gain                                                          -                   (54)
   Provisions for contract, restructuring, inventory and warranty reserves               30                    56
Changes in assets and liabilities                                                 
   Accounts receivable                                                               (2,222)               (1,577)
   Unbilled costs and fees                                                              741                   (36)
   Inventories                                                                           99                   214
   Prepaid expenses and other current assets                                           (155)                   15
   Accounts payable                                                                    (316)                  862
   Customer deposits                                                                     20                  (132)
   Accrued expenses and other current liabilities                                       392                  (575)
   Accrued restructuring costs                                                            -                  (245)
   Noncurrent liabilities                                                                 7                    17
                                                                                    -------               -------
Total adjustments                                                                    (1,434)               (1,257)
                                                                                    -------               -------

Net cash used for operating activities                                                 (975)               (1,658)
                                                                                    -------               -------

CASH FLOWS FROM INVESTING ACTIVITIES:
   (Purchase)/sale of property and equipment                                             33                  (340)
                                                                                    -------               -------
   Cash provided by (used for) investing activities                                      33                  (340)
                                                                                    -------               -------

CASH FLOWS FROM FINANCING ACTIVITIES:
   Proceeds from borrowings                                                             650                     -
   Repayment of officer note                                                           (650)                    -
   Proceeds from exercise of stock options                                               40                    14
   Proceeds from issuance of stock                                                      902                     -
   Principal payments of capital lease obligations                                     (156)                 (116)
                                                                                    -------               -------
   Cash (used for) provided by financing activities                                     786                  (102)
                                                                                    -------               -------
                                                                            
NET DECREASE IN CASH AND CASH EQUIVALENTS                                              (156)               (2,100)
CASH AND CASH EQUIVALENTS AT BEGINNING OF PERIOD                                        869                 2,496
                                                                                    -------               -------
CASH AND CASH EQUIVALENTS AT END OF PERIOD                                          $   713               $   396
                                                                                    =======               =======
                                                                              
SUPPLEMENTAL DISCLOSURES OF CASH FLOW INFORMATION:
   Interest paid                                                                    $    47               $     9
   Income taxes paid                                                                $     -               $     -


</TABLE>

         The accompanying notes are an integral part of these condensed 
         consolidated financial statements.



                                      -5-
<PAGE>   6

              
                    AMERICAN SCIENCE AND ENGINEERING, INC.

          PREPARATION OF CONDENSED CONSOLIDATED FINANCIAL STATEMENTS


The condensed consolidated financial statements included herein have
been prepared by American Science and Engineering, Inc. (the Company) pursuant
to the rules and regulations of the Securities and Exchange Commission and the
annual condensed consolidated financial statements are subject to year end audit
by independent public accountants.  Certain information and footnote disclosures
normally included in financial statements prepared in accordance with generally
accepted accounting principles have been condensed or omitted pursuant to such
rules and regulations.  The Company believes,however, that the disclosures are
adequate to make the information presented not misleading.  It is suggested that
these condensed consolidated financial statements be read in conjunction with
the consolidated financial statements and the notes thereto included in the
Company's latest annual report on Form 10-K.

The condensed consolidated financial statements, in the opinion of
management, include all adjustments necessary to present fairly the Company's
financial position and the results of operations.  Certain accounts have been
reclassified to conform with the current year's presentation.  See Management's
Discussion and Analysis of Financial Condition and Results of Operations on page
8.  These results are not necessarily to be considered indicative of the results
for the entire year.

The condensed consolidated financial statements contained herein have been 
reviewed by Arthur Andersen LLP, the Company's independent public accountants 
(see attached letter on page 11).


NOTES TO CONDENSED CONSOLIDATED FINANCIAL STATEMENTS (Unaudited)

1.   Market and Operating Environment
     --------------------------------
     As described in its latest Form 10-K, the Company sustained significant 
     losses in the previous two years.  While management has succeeded in 
     eliminating these losses during the current fiscal year, the Company's 
     liquidity and cash reserves were depleted to a point where outside 
     financing was required to fund the growth of operations back to the point
     of cash sufficiency.

     Accordingly, several outside financing initiatives were accomplished
     during the first nine months of fiscal 1996, the sum of which produced a
     cash infusion of $1.5 million and an additional $4 million of bank
     borrowing potential based on the level of domestic and export business
     attained.

     With these initiatives, management believes that sufficient liquidity
     has been attained to sustain operations.  However, revenues could continue
     to be volatile, new cash pressures may arise, and additional outside
     financing may not be immediately available. 

2.   Significant Accounting Policies
     -------------------------------

     The significant accounting policies followed by the Company and its
     subsidiary in preparing its consolidated financial statements are set forth
     in Note 2 to the consolidated financial statements included in Form 10-K
     for the year ended March 31, 1995.  The Company has made no change in these
     policies during this quarter.

                                     -6-

<PAGE>   7

<TABLE>

3.   Inventories
     ------------       
<CAPTION>

     Inventories consisted of:          December 29, 1995     March 31, 1995
                                        -----------------     --------------
        <S>                                    <C>                <C>
        Raw materials and completed         
          sub-assemblies                       $2,386,000         $2,485,000
        Work in process                         1,224,000          1,224,000
                                               ----------         ----------
        Total                                  $3,610,000         $3,709,000
                                               ==========         ==========
</TABLE>
                                                                   
                                                                   
4.   Income (Loss) per Common and Common Equivalent Share
     ----------------------------------------------------

     Income (loss) per common share for the three and nine month periods
     ended December 29, 1995 and December 30, 1994 was computed by dividing net
     income by the weighted average number of common stock and common stock
     equivalents outstanding during the year.    Stock options and stock
     warrants were considered in the computation of earnings per common and
     common equivalent shares.  Stock options and stock warrants were not
     considered in computing the loss per share as the effect would have been
     anti-dilutive.
                                                                   
5.   Income Taxes
     ------------

     The Company provides for income taxes on an interim basis based on the
     estimated effective tax rate for the fiscal year.  No income tax provision
     has been recorded in connection with the first nine months' income, as the
     Company has significant net operating loss carryforwards.  Such
     carryforwards, which are subject to review by the Internal Revenue Service,
     approximate $9,817,000 at March 31, 1995, and expire beginning in 1999. 
     The Company also has unused investment tax and other credits of
     approximately $318,000 expiring from 1996 through 2001.


                                     -7-
<PAGE>   8

              

                    AMERICAN SCIENCE AND ENGINEERING, INC.
          ITEM 2 - MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL
                       CONDITION AND RESULTS OF OPERATIONS

Overview 
--------
For the past ten quarters, management has pursued an aggressive turnaround 
program to rebuild sales, control costs, and return the Company to sustainable
profitability.  In the third fiscal quarter ended December 29, 1995, continued
progress was made in all such areas.

Net sales and contract revenues grew to $4,717,000 during the quarter, a
22% increase over the previous  second quarter of fiscal 1996 and a 17% increase
from the comparable year ago period.  A net profit of $252,000 was reported in
the current quarter compared to a profit of $155,000 in the previous quarter and
net income of $61,000 in the corresponding year ago period.

Finally, continued progress was made in improving the Company's liquidity and 
cash position through a significant outside financing transaction which will 
be discussed further in this section.

Backlog
-------
Total backlog at the end of the third quarter was approximately the same
as reported at September 29, 1995.  New bookings for X-ray inspection systems
included orders from the U.S. State Department, the U.S. Navy, the Federal
Deposit Insurance Corp., numerous commercial customers, and several
international agencies.

Results of Operations
---------------------
Net sales and contract revenues in the third quarter increased by $697,000 
(17%) compared to the corresponding period a year ago and increased by 
$835,000 (22%) compared to the second quarter of fiscal year 1996.  Compared to
the previous quarter, security systems and related field service revenues were
up by $709,000 (21%), research and engineering revenues were up by $152,000
(28%), and control systems revenues were down $26,000 (79%).

One positive development during the current quarter was the completion,
successful testing and delivery of the Company's first mobile, truck-mounted
version of the CargoSearch [TRADEMARK] X-ray inspection system.  This contract
was jointly funded by the U.S. Customs Service and the Defense Department's
Advanced Research Projects Agency (ARPA).   MobileSearch  [TRADEMARK] allows
fast deployment and surprise searches of cargo and vehicles for drugs,
contraband and explosives.

Additionally, the Company continues to gain commercial marketplace
acceptance as companies recognize the need to improve their building security. 
Orders booked during the quarter included sales to eight new corporate accounts.

For the third quarter, costs of sales and contracts increased to
$3,297,000 from $2,685,000 compared to the corresponding period a year ago due
primarily to increased sales volume of X-ray inspection systems and field
services revenues in the current period.  Costs of sales and contracts
represented 70% of revenues versus 67% for the corresponding period last year
and 66% for the previous second quarter of fiscal year 1996.

The costs of sales percentage of revenues in the current quarter
increased over the previous quarter primarily due to the sales mix of lower
margin products partially offset by higher overhead absorption resulting from
increased product shipments and reduced occupancy costs from the new facility.

                                     -8-

<PAGE>   9

Selling, general and administrative expenses of $1,030,000 for the third
quarter were lower by 7% compared to the corresponding year-ago period and
higher by 7% compared to the second quarter of fiscal 1996.  As a percent of
sales, selling, general and administrative expenses were 22% of revenues in the
current quarter compared to 28% of revenues for the corresponding year-ago
period and 25% for the second quarter of fiscal year 1996.  This declining ratio
of SG&A expenses to sales is primarily the result of higher revenues and the
cessation of non-recurring legal and relocation expenses that occurred in
previous quarters.

Company-funded research and development expenses of $113,000 for the
third quarter were lower by $68,000 (38%) compared to the year-ago quarter and
lower by $15,000 (12%) compared to the second quarter of fiscal year 1996.

The Company produced a net profit of $252,000 during the third quarter. 
This is an improvement of $191,000 over net income in the year-ago quarter and
an improvement of $97,000 over the net profit reported in the previous second
quarter of fiscal 1996.  The improved profitability results from higher revenue,
reduced overhead levels associated with the new facility, and controlled SG&A
spending.

Liquidity and Capital Resources
-------------------------------

The Company continued to improve its liquidity and capital resources
during and subsequent to the current period.  Net cash used for operating
activities during the first nine months of fiscal year 1996 was $975,000,
compared to $1,658,000 net cash used during the corresponding year-ago period. 
Cash and cash equivalents were largely unchanged from both September 29, 1995
and March 31, 1995.  Working capital (current assets minus current liabilities)
increased by $1,528,000 (33%) since March 31, 1995, growing from $4,616,000 to
$6,144,000 at the end of the third quarter.

On January 11, 1996, a domestic line of credit for $1.5 million was
closed with a local bank.  This line of credit is secured by accounts receivable
from ongoing U.S. customers.  In January, borrowings of $400,000 under this new
line were approved.  Also on January 11, 1996, an additional $1.5 milllion
borrowing facility was closed to finance certain short lead-time orders the
Company anticipates receiving over the next few months.

This above-mentioned new financing agreement is in addition to the
existing  $1 million line of credit agreement secured by foreign accounts
receivable and in-process inventory for export orders and guaranteed by the U.S.
Export Import Bank.

These new sources of outside capital, together with the improvements in
operating cash flow, have relieved the acute cash shortages of previous quarters
and provide a base of financing for future operations.


                                     -9-


<PAGE>   10

               
                    AMERICAN SCIENCE AND ENGINEERING, INC.

Part II - Other Information

<TABLE>
Item 6.   Exhibits and Reports on Form 8-K
          --------------------------------

(a)           Exhibit Index
              -------------
<CAPTION>
                        Exhibit                         Page
                        -------                         ----
         <S>                                             <C>
         (11) Statement re:  Computation of               7
              Income (Loss) per Common
              and Common Equivalent Share                

         (28) Letter Concerning Review by                11
              Independent Public Accountant            

(b)           Reports on Form 8-K
              -------------------
</TABLE>

          The following reports on Form 8-K were filed during the quarter.

              NONE

              SIGNATURES
              ----------

Pursuant to the requirements of the Securities Exchange Act of 1934, the
Registrant has duly caused this Report to be signed on its behalf by the
undersigned hereunto duly authorized.


                                  AMERICAN SCIENCE AND ENGINEERING, INC.
                                  (Registrant)


Date:  30 January 1996                ------------------------------------------
                                      Lee C. Steele
                                      Vice President and Chief Financial Officer

                                     -10-

<PAGE>   11

                        

                             ARTHUR ANDERSEN LLP
                                  
                   REPORT OF INDEPENDENT PUBLIC ACCOUNTANTS
                   ----------------------------------------

To the Board of Directors of
American Science and Engineering, Inc.:


We have reviewed the accompanying condensed consolidated balance sheet
of American Science and Engineering, Inc. (a Massachusetts corporation) and
subsidiary as of December 29, 1995, and the related condensed consolidated
statements of operations for the three months and nine months then ended and
cash flows for the nine months then ended.  These condensed consolidated
financial statements are the responsibility of the Company's management.

We conducted our review in accordance with standards established by the
American Institute of Certified Public Accountants.  A review of interim
financial information consists principally of applying analytical procedures to
financial data and making inquiries of persons responsible for financial and
accounting matters.  It is substantially less in scope than an audit conducted
in accordance with generally accepted auditing standards, the objective of which
is the expression of an opinion regarding the financial statements taken as a
whole.  Accordingly, we do not express such an opinion.

Based on our review, we are not aware of any material modifications that
should be made to the condensed consolidated financial statements referred to
above for them to be in conformity with generally accepted accounting
principles.



                              /s/Arthur Andersen LLP



Boston, Massachusetts
January 26, 1996



                                     -11-