<PAGE>   1
                                   FORM 10-Q
                       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 31, 1997
                                               -----------------

                                       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)

                                 (978) 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                             Dec. 31, 1997
      ---------------------                             -------------
      $.66 2/3 par value                                  4,724,650


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


<PAGE>   2
                     AMERICAN SCIENCE AND ENGINEERING, INC.
                         PART I - FINANCIAL INFORMATION
                 CONDENSED CONSOLIDATED STATEMENTS OF OPERATIONS
                                   (Unaudited)

<TABLE>
<CAPTION>
                                                               For The Three Months Ended     For The Nine Months Ended
                                                               --------------------------     -------------------------
Dollars and shares in Thousands, except per share amounts         Dec. 31,     Dec. 27,          Dec. 31,     Dec. 27,  
                                                                    1997         1996              1997         1996
                                                                  -------      -------            -------      -------
                                                                                                
<S>                                                               <C>          <C>                <C>          <C>     
NET SALES AND CONTRACT REVENUE                                    $ 7,900      $ 7,127            $24,137      $20,511 
                                                                  -------      -------            -------      -------
COSTS AND EXPENSES:                                                                            
   Cost of sales and contracts                                      4,692        4,295             14,658       13,232
   Selling, general and administrative expenses                     1,883        1,779              5,516        4,771
   Research and development                                           569          538              1,914        1,132
                                                                  -------      -------            -------      -------
   Total costs and expenses                                         7,144        6,612             22,088       19,135
                                                                  -------      -------            -------      -------

OPERATING INCOME                                                      756          515              2,049        1,376
                                                                  -------      -------            -------      -------
OTHER INCOME (EXPENSE):                                                                         
   Interest, net                                                       30           18                100           88
   Other, net                                                          (5)          (9)               (16)         (44)
                                                                  -------      -------            -------      -------
   Total other income                                                  25            9                 84           44
                                                                  -------      -------            -------      -------
                                                                                                
INCOME BEFORE PROVISION                                                                         
FOR INCOME TAXES                                                      781          524              2,133        1,420
                                                                                                
PROVISION FOR INCOME TAXES                                             30           21                 87           60
                                                                  -------      -------            -------      -------
                                                                                                
NET INCOME                                                        $   751      $   503            $ 2,046      $ 1,360
                                                                  =======      =======            =======      =======
                                                                                                
INCOME PER SHARE - PRIMARY/FULLY DILUTED                          $   .15      $   .10            $   .42      $   .28
                                                                  =======      =======            =======      =======

DIVIDENDS PAID PER SHARE                                             NONE         NONE               NONE         NONE
                                                                  =======      =======            =======      =======
                                                                                                
WEIGHTED AVERAGE SHARES  - PRIMARY                                  4,930        4,822              4,869        4,801
                                                                  =======      =======            =======      =======
                         - FULLY                                                                
                           DILUTED                                  5,013        4,846              4,925        4,816
                                                                  =======      =======            =======      =======
</TABLE>

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


                                       -2-

<PAGE>   3

                     AMERICAN SCIENCE AND ENGINEERING, INC.
                      CONDENSED CONSOLIDATED BALANCE SHEETS

<TABLE>
<CAPTION>
Dollars in Thousands                                   Dec. 31, 1997     Mar. 28, 1997
                                                       -------------     -------------
                                                       (Unaudited)
<S>                                                     <C>                 <C>    
ASSETS    CURRENT ASSETS:
          Cash and cash equivalents                     $ 2,148             $ 3,202
          Accounts receivable, net                        7,272               5,019
          Unbilled costs and fees, net                    2,306                 981
          Inventories                                     6,297               4,736
          Prepaid expenses and other current assets          90                 157
                                                        -------             -------
          TOTAL CURRENT ASSETS                           18,113              14,095
                                                        -------             -------
          NON CURRENT ASSETS:                                             
          Deposits                                           24                 115
          Property and equipment, net of                                  
            accumulated depreciation of                                   
            $9,306 at Dec. 31, 1997 and                                   
            $8,860 at March 28, 1997                      1,769               1,304
                                                        -------             -------
                                                                          
                                                        $19,906             $15,514
                                                        =======             =======
</TABLE>

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


                                       -3-

<PAGE>   4

                     AMERICAN SCIENCE AND ENGINEERING, INC.
                      CONDENSED CONSOLIDATED BALANCE SHEETS
                                   (CONTINUED)

<TABLE>
<CAPTION>
Dollars in Thousands                                         Dec. 31, 1997    Mar. 28, 1997
                                                             -------------    -------------
                                                              (Unaudited)
<S>                                                             <C>             <C>    
LIABILITIES &   CURRENT LIABILITIES:
STOCKHOLDERS'   Current maturities of obligations
INVESTMENT       under capital leases                           $    19         $    18
                Accounts payable                                  2,713           2,253
                Accrued salaries and benefits                       654             573
                Accrued commissions                                 354              82
                Accrued warranty costs                              566             292
                Deferred revenue                                  1,177             526
                Other current liabilities                           463             495
                                                                -------         -------
                TOTAL CURRENT LIABILITIES                         5,946           4,239
                                                                -------         -------
                NONCURRENT LIABILITIES:
                Obligations under capital leases, net
                  of current maturities                              27              42
                Deferred Revenue                                    424             660
                Deferred compensation                               167             174
                Deferred rent                                       274             249
                                                                -------         -------
                TOTAL NONCURRENT LIABILITIES                        892           1,125
                                                                -------         -------
                STOCKHOLDERS' INVESTMENT:
                Preferred stock, no par value
                Authorized - 100,000 shares
                Issued - None
                Common stock, $.66-2/3 par value
                Authorized - 20,000,000 shares
                Issued 4,724,650 shares at Dec. 31, 1997
                and 4,585,209 shares at Mar. 28, 1997             3,150           3,058
                Capital in excess of par value                   16,072          15,273
                Accumulated deficit                              (5,319)         (7,365)
                                                                -------         -------
                                                                 13,903          10,966
                Note receivable-Officer                            (640)           (640)
                Less: treasury stock -
                64,907 shares at Dec. 31, 1997 and
                  62,481 shares at March 28, 1997,
                  at cost, respectively                            (195)           (176)
                                                                -------         -------

                TOTAL STOCKHOLDERS' INVESTMENT                   13,068          10,150
                                                                -------         -------
                                                                $19,906         $15,514
                                                                =======         =======

</TABLE>

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


                                       -4-

<PAGE>   5

                     AMERICAN SCIENCE AND ENGINEERING, INC.
                 CONDENSED CONSOLIDATED STATEMENTS OF CASH FLOWS
                                   (Unaudited)

<TABLE>
<CAPTION>
Dollars in Thousands                                                 For the Nine Months Ended
                                                                  ------------------------------
                                                                  Dec. 31, 1997    Dec. 28, 1996
                                                                  -------------    -------------
<S>                                                                  <C>             <C>    
CASH FLOWS FROM OPERATING ACTIVITIES:
Net income                                                           $ 2,046         $ 1,360
Adjustments to reconcile net income to net cash (used for)
operating activities:
  Depreciation and amortization                                          446             254
  Provisions for contract, restructuring, inventory, accounts
  receivable and warranty reserves                                       777             723
Changes in assets and liabilities:
  Accounts receivable                                                 (2,253)         (2,339)
  Unbilled costs and fees                                             (1,375)            443
  Inventories                                                         (1,761)           (206)
  Prepaid expenses and other assets                                      158             267
  Accounts payable                                                       460             288
  Customer deposits                                                       --          (2,396)
  Accrued expenses and other current liabilities                         852             679
  Noncurrent liabilities                                                (218)              5
                                                                     -------         -------
  Total adjustments                                                   (2,914)         (2,282)
                                                                     -------         -------

Net cash used for operating activities                                  (868)           (922)
                                                                     -------         -------
CASH FLOWS FROM INVESTING ACTIVITIES:
  Purchase of property and equipment                                    (911)           (561)
                                                                     -------         -------
CASH FLOWS FROM FINANCING ACTIVITIES:
  Proceeds from exercise of stock options                                739             232
  Principal payments of capital lease obligations                        (14)            (11)
                                                                     -------         -------
  Cash provided by financing activities                                  725             221
                                                                     -------         -------

NET DECREASE IN CASH AND CASH EQUIVALENTS                             (1,054)         (1,262)
CASH AND CASH EQUIVALENTS AT BEGINNING OF PERIOD                       3,202           3,377
                                                                     -------         -------
CASH AND CASH EQUIVALENTS AT END OF PERIOD                           $ 2,148         $ 2,115
                                                                     =======         =======
SUPPLEMENTAL DISCLOSURES OF CASH FLOW
INFORMATION:
  Interest paid                                                      $     5         $     8
  Income taxes paid                                                  $    53         $    30

NON-CASH TRANSACTIONS:
  Issuance of stock in lieu of fees                                  $   133         $   258
</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. These results are not necessarily to be
considered indicative of the results for the entire year.


NOTES TO CONDENSED CONSOLIDATED FINANCIAL STATEMENTS (Unaudited)

1.   NATURE OF OPERATIONS AND SIGNIFICANT ACCOUNTING POLICIES
     American Science and Engineering, Inc., is engaged in the development and
     manufacture of sophisticated X-ray inspection systems for critical
     detection and security screening solutions for sale primarily to U.S. and
     foreign government agencies.

     The significant accounting policies followed by the Company and its
     subsidiary in preparing its consolidated financial statements are set forth
     in Note 1 to the consolidated financial statements included in Form 10-K
     for the year ended March 28, 1997.

     Effective quarter ended September 30, 1997, the Company has elected to
     change financial reporting from a fiscal month end to a calender month end.
     In addition, the year end reporting period will end on March 31, 1998. This
     change in month end has no material effect on the results of operations for
     the quarter ended December 31, 1997.

2.   INVENTORIES 
     Inventories consisted of:

<TABLE>
<CAPTION>
                                            Dec. 31, 1997     Mar. 28, 1997
                                            -------------     -------------
          <S>                                <C>               <C>       
          Raw materials and completed
           subassemblies                     $4,075,000        $3,106,000
          Work in process                     2,122,000         1,334,000
          Finished Goods                        100,000           296,000
                                             ----------        ----------
          Total                              $6,297,000        $4,736,000
                                             ==========        ==========
</TABLE>

3.   INCOME PER COMMON AND COMMON EQUIVALENT SHARE
     Income per common share for the nine-month period ended December 31, 1997,
     was computed by dividing net income by the weighted average number of
     common stock and common stock equivalents outstanding during the year.
     Dilutive stock options and stock warrants were considered in the
     computation of earnings per common and common equivalent shares.

                                       -6-

<PAGE>   7

                     AMERICAN SCIENCE AND ENGINEERING, INC.
           PREPARATION OF CONDENSED CONSOLIDATED FINANCIAL STATEMENTS
                                   (Continued)


4.   INCOME TAXES
     At March 28, 1997, the Company had approximately $6,986,000 of federal net
     operating loss carryforwards which can be used, subject to certain
     limitations, to offset future federal taxable income. The carryforwards
     expire through the year 2010. The Company also has unused investment tax
     and other credits of approximately $235,000 expiring through 2001. The
     provision for income taxes in the accompanying statement of income is
     substantially a current provision and differs from the provision calculated
     by applying the statutory federal income tax rate of 34% to income before
     provision for income taxes due to the following:


<TABLE>
<CAPTION>
                                                          Nine Months Ended
                                                            Dec. 31, 1997
                                                          -----------------
          <S>                                                 <C>      
          Provision for income taxes at statutory rate        $ 725,000
          Benefit from net operating loss carryforward         (725,000)
          Other tax liabilities                                  87,000
                                                              ---------
                                                              $  87,000
                                                              =========
</TABLE>


                                       -7-

<PAGE>   8



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


OVERVIEW
In the third quarter of fiscal 1998, the Company produced its eleventh
consecutive quarter of increasing profits. Net sales and contract revenues were
$7,900,000 during the quarter, an 11% increase from the comparable year ago
period and a 9% decrease from the second quarter of fiscal 1998. The Company
earned net profits of $751,000 in the current quarter, a 49% increase compared
to net profits of $503,000 in the corresponding year ago period and a 10%
increase compared to net profits of $685,000 in the previous quarter.

RESULTS OF OPERATIONS
Net sales and contract revenues of $7,900,000 in the third quarter increased by
$773,000 (11%) in comparison to the corresponding period a year ago and
decreased by $805,000 (9%) compared to the second quarter of fiscal year 1998.
Compared to the previous quarter, security systems and related field service
revenues were lower by $493,000 (6%) due to the timing of orders now expected to
be received and shipped during the fourth quarter and beyond. Research and
engineering revenues were lower by $312,000 (39%) as compared to the second
quarter. This quarter to quarter decrease in research and engineering revenues
is due to slower than anticipated start-up of a $3.8 million contract for two
additional prototype MobileSearch(TM) systems during the third quarter.

For the third quarter, costs of sales and contracts increased to $4,692,000 from
$4,295,000 in the corresponding period a year ago primarily due to increased
sales volume of X-ray inspection systems and field service revenues in the
current period. Costs of sales and contracts represented 59.4% of revenues
versus 60.3% for the corresponding period last year and 59.6% for the second
quarter of fiscal year 1998.

Selling, general and administrative expenses of $1,883,000 for the third quarter
were higher by 6% compared to the corresponding year-ago period and lower by 4%
compared to the second quarter of fiscal 1998. As a percent of sales, selling,
general and administrative expenses were 24% of revenues in the current quarter
compared to 25% and 23% of revenues for the corresponding year-ago period and
the second quarter of fiscal year 1998, respectively.

Company-funded research and development expenses of $569,000 for the third
quarter were higher by $31,000 (6%) compared to the year-ago quarter and lower
by $312,000 (35%) compared to the second quarter of fiscal year 1998. This
decrease in quarter to quarter research and development expenditures is due to
the timing of certain development expenditures. The Company remains committed to
increasing the level of research and development activities in subsequent
quarters.

The Company produced a net profit of $751,000 during the third quarter. This is
an improvement of $248,000 (49%) over net profit in the year-ago quarter and an
improvement of $66,000 (10%) over the net profit reported in the second quarter
of fiscal 1998. The quarter to quarter improvement in profitability results
primarily from lower selling, general and administrative expenses and Company
funded research and development.

                                       -8-

<PAGE>   9



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


LIQUIDITY AND CAPITAL RESOURCES
Net cash used by operating activities during the first nine months of fiscal
year 1998 was $868,000, compared to $922,000 net cash used during the
corresponding year-ago period. Cash and cash equivalents decreased by $1,054,000
to $2,148,000 as of December 31, 1997, compared to $3,202,000 on March 28, 1997.
This decrease is largely due to increases in accounts receivable, unbilled costs
and fees, inventory, as well as property, plant and equipment. Working capital
increased by $2,311,000 (23%) since March 28, 1997, growing from $9,856,000 to
$12,167,000 at the end of the third quarter.

No external borrowings were outstanding at the end of either the current or
previous quarter. Management believes the Company has access to sufficient
capital resources to support operations over the next several quarters.


                                       -9-

<PAGE>   10


                     AMERICAN SCIENCE AND ENGINEERING, INC.

Part II  - Other Information

Item 1. Legal Proceedings
In the Company's lawsuit against Martin Annis and AnnisTech, Inc., the
Massachusetts Superior Court issued a preliminary injunction prohibiting Dr.
Annis and AnnisTech from using AS&E's trade secrets or proprietary technology
and from working on a particular government contract. The injunction was upheld
by a single justice of the Massachusetts Appeals Court and remains in effect as
of the date of this filing.

On January 28, 1998, the United States District Court for the District of
Massachusetts granted Vivid Technologies, Inc.'s request for a declaratory
judgment that Vivid does not infringe on two claims of the Company's U.S.
Patent No. 5,253,283. The Company expects to appeal the judgment on several
grounds.

Item 6. Exhibits and Reports on Form 8-K
        --------------------------------

          (a)  EXHIBIT INDEX
               Exhibit 27 - Financial Data Schedule

          (b)  REPORTS ON FORM 8-K
               No reports on Form 8-K were filed during the quarter.

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: 13 February 1998              
                                      ------------------------------------------
                                      Lee C. Steele
                                      Vice President and Chief Financial Officer


SAFE HARBOR STATEMENT
The foregoing 10-Q contains statements concerning the Company's financial
performance and business operations which may be considered "forward-looking"
under applicable securities laws.

The Company wishes to caution readers of this Form 10-Q that actual results
might differ materially from those projected in any forward-looking statements.

Factors which might cause actual results to differ materially from those
projected in the forward-looking statements contained herein include the
following: Significant reductions or delays in procurements of the Company's
systems by the United States Government; disruption in the supply of any
sole-source component incorporated into the Company's products (of which there
are several); litigation seeking to restrict the use of intellectual property
used by the Company; potential product liability claims against the Company;
global political trends and events which affect public perception of the threat
presented by drugs, explosives and other contraband; the ability of governments
and private organizations to fund purchases of the Company's products to address
such threats; and the potential insufficiency of Company resources, including
human resources, capital, plant and equipment and management systems, to
accommodate any future growth. These and certain other factors which might cause
actual results to differ materially from those projected are more fully set
forth under the caption "Risk Factors" in the Company's Registration Statement
on Form S-3 (SEC File No. 333-9151).

                                      -10-