<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 June 30, 1998
                                                 -------------

                                       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                                         June 30, 1998
----------------------                                        --------------
 $.66 2/3 par value                                             4,771,800




                               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
                                                            ------------------------------
Dollars and shares in thousands, except per share amounts   June 30, 1998    June 27, 1997
                                                            -------------    -------------
<S>                                                             <C>             <C>          

NET SALES AND CONTRACT REVENUE                                  $11,692         $7,532       
                                                                -------         ------
                                                                                       
COSTS AND EXPENSES:                                                                    
   Cost of sales and contracts                                    7,239          4,782 
   Selling, general and administrative expenses                   2,131          1,670 
   Research and development                                       1,408            464 
                                                                -------         ------
   Total costs and expenses                                      10,778          6,916 
                                                                -------         ------
                                                                                       
OPERATING INCOME                                                    914            616 
                                                                -------         ------
                                                                                       
OTHER INCOME (EXPENSE):                                                                
   Interest, net                                                     10             27 
   Other, net                                                       (48)            (3)
                                                                -------         ------
   Total other income (expense)                                     (38)            24 
                                                                -------         ------
                                                                                       
INCOME BEFORE PROVISION FOR INCOME TAXES                            876            640 

PROVISION FOR INCOME TAXES                                          350             30   
                                                                -------         ------
                                                                                    
NET INCOME                                                      $   526         $  610 
                                                                =======         ====== 

INCOME PER SHARE - BASIC                                        $   .11         $  .13
                                                                =======         ====== 
                 - DILUTED                                      $   .10         $  .13 
                                                                =======         ======
                                                                                    
DIVIDENDS PAID PER SHARE                                           None           None
                                                                =======         ====== 
 
WEIGHTED AVERAGE SHARES - BASIC                                   4,749          4,562 
                                                                =======         ======
                        - DILUTED                                 5,111          4,805
                                                                =======         ======
                                                                      
</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
                                                           June 30, 1998      March 31, 1998
                                                           -------------      --------------
                                                             (Unaudited)
<S>                                                           <C>                 <C>    

 ASSETS     CURRENT ASSETS:
            Cash and cash equivalents                         $ 1,401             $ 2,290
            Accounts receivable, net                            7,760               6,955
            Unbilled costs and fees, net                        4,955               3,190
            Inventories                                         9,396               8,737
            Deferred income taxes                               2,094               2,351
            Prepaid expenses and other current assets             764                 389
                                                              -------             -------
            TOTAL CURRENT ASSETS                               26,370              23,912
                                                              -------             -------

            NON CURRENT ASSETS:
            Non-current deferred income taxes                     205                 205
            Deposits                                               16                  24
            Property and equipment, net
               of accumulated depreciation of
               $9,578 at June 30, 1998 and
               $9,394 at March 31, 1998                         2,218               1,852
                                                              -------             -------
                                                              $28,809             $25,993
                                                              =======             =======


</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
                                                                      June 30, 1998        March 31, 1998
                                                                      -------------        --------------
                                                                       (Unaudited)
<S>                  <C>                                                 <C>                  <C>

LIABILITIES &       CURRENT LIABILITIES:
STOCKHOLDERS'       Line of credit                                       $ 1,000              $    --
INVESTMENT          Current maturities of obligations
                       under capital leases                                   20                   20
                    Accounts payable                                       4,387                4,360
                    Accrued salaries and benefits                          1,026                  831
                    Accrued warranty costs                                   446                  497
                    Accrued income taxes                                     375                  615
                    Deferred revenue                                         992                1,240
                    Customer deposits                                      2,531                1,151
                    Other current liabilities                                477                  507
                                                                         -------              -------
                    TOTAL CURRENT LIABILITIES                             11,254                9,221
                                                                         -------              -------

                    NONCURRENT LIABILITIES:
                    Obligations under capital leases, net
                       of current maturities                                  17                   22
                    Deferred revenue                                         243                  232
                    Deferred compensation                                    154                  154
                    Deferred rent                                            281                  280
                                                                         -------              -------
                    TOTAL NONCURRENT LIABILITIES                             695                  688
                                                                         -------              -------

                    STOCKHOLDERS' INVESTMENT:
                    Jr. Preferred stock, no par value
                    Authorized - 10,000 shares, issued none

                    Preferred stock, no par value
                    Authorized - 100,000 shares, issued none

                    Common stock, $.66-2/3 par value
                    Authorized -20,000,000 shares
                    Issued 4,771,800 shares at June 30, 1998
                     and 4,743,569 shares at March 31, 1998                3,181                3,162
                    Capital in excess of par value                        16,509               16,278
                    Accumulated deficit                                   (2,178)              (2,704)
                                                                         -------              -------
                                                                          17,512               16,736
                    Note receivable-Officer                                 (640)                (640)
                                                                      
                    Less: treasury stock -
                    6,678 shares at June 30, 1998
                      and March 31, 1998 at cost                             (12)                 (12)
                                                                         -------              -------
                                                                      
                    TOTAL STOCKHOLDERS' INVESTMENT                        16,860               16,084
                                                                         -------              -------
                                                                         $28,809              $25,993
                                                                         =======              =======


</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 Three Months Ended
                                                                        --------------------------------
                                                                        June 30, 1998      June 27, 1997    
                                                                        -------------      -------------    
<S>                                                                        <C>               <C>            
                                                                                                            
CASH FLOWS FROM OPERATING ACTIVITIES:                                                                       
Net income                                                                 $   526           $   610        
Adjustments to reconcile net income to net cash provided by                                                 
(used for) operating activities:                                                                            
   Depreciation and amortization                                               184               152        
   Provisions for contract, inventory and warranty reserves                    257               208        
                                                                                                            
Changes in assets and liabilities:                                                                          
   Accounts receivable                                                        (805)           (1,698)       
   Unbilled costs and fees                                                  (1,765)             (974)       
   Inventories                                                                (659)             (332)       
   Prepaid expenses, other current assets, and deposits                       (367)               58        
   Accounts payable                                                             27               432        
   Customer deposits                                                         1,380               137        
   Accrued expenses and other current liabilities                             (374)              438        
   Noncurrent liabilities                                                       12                53        
                                                                           -------           -------        
Total adjustments                                                           (2,110)           (1,526)       
                                                                           -------           -------        
                                                                                                            
Net cash used for operating activities                                      (1,584)             (916)       
                                                                           -------           -------        
                                                                                                            
CASH FLOWS FROM INVESTING ACTIVITIES:                                                                       
   Purchase of property and equipment, net                                    (550)             (340)       
                                                                           -------           -------        
                                                                                                            
CASH FLOWS FROM FINANCING ACTIVITIES:                                                                       
   Proceeds from exercise of stock options                                     250               378        
   Net borrowings Line of credit                                             1,000                --        
   Principal payments of capital lease obligations                              (5)               (5)       
                                                                           -------           -------        
   Cash provided by financing activities                                     1,245               373        
                                                                           -------           -------        
                                                                                                            
NET INCREASE IN CASH AND CASH EQUIVALENTS                                     (889)             (883)       
CASH AND CASH EQUIVALENTS AT BEGINNING OF PERIOD                             2,290             3,202        
                                                                           -------           -------        
CASH AND CASH EQUIVALENTS AT END OF PERIOD                                 $ 1,401           $ 2,319        
                                                                           =======           =======        
                                                                                                            
SUPPLEMENTAL DISCLOSURES OF CASH FLOW INFORMATION:                                                          
   Interest paid                                                           $     9           $     2        
   Income taxes paid                                                       $   324           $    --        
                                                                                                            
</TABLE>
                                                                           



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




                                      -5-
<PAGE>   6

                     AMERICAN SCIENCE AND ENGINEERING, INC.
                   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 31, 1998. The Company has made no change in
      these policies during this quarter.


2.    INVENTORIES
      Inventories consisted of: (dollars in thousands)


<TABLE>
<CAPTION>
                                             June 30, 1998       March 31,1998
                                             -------------       -------------
          <S>                                    <C>                 <C>

          Raw materials and completed
             sub-assemblies                      $4,990              $4,958
          Work in process                         4,281               3,654
          Finished goods                            125                 125
                                                 ------              ------ 
          Total                                  $9,396              $8,737
                                                 ======              ======
</TABLE>




                                      -6-
<PAGE>   7


3.    INCOME PER COMMON AND COMMON EQUIVALENT SHARE
      In March 1997, the Financial Accounting Standards Board (FASB) issued
      statement of Financial Accounting Standards (SFAS) No. 128, `Earnings per
      Share"," which establishes standards for computing and presenting earnings
      per share for entities with publicly held common stock or potential common
      stock. The Company adopted SFAS 128 in fiscal 1998 and as required,
      restated per share amounts for all prior periods presented to conform to
      the new requirements. Basic earnings per common share is computed by
      dividing net income by the weighted average number of shares of common
      stock outstanding during the year. No dilution for any potentially
      dilutive securities is included. Diluted earnings per share includes the
      dilutive impact of options and warrants using the average share price of
      the Company's common stock for the period.

<TABLE>
<CAPTION>

      Earnings Per Share                                      Three Months Ended
      ------------------                                 -----------------------------
      In thousands except per share amounts              June 30, 1998   June 27, 1997
                                                         -------------   -------------
      <S>                                                   <C>              <C>

      BASIC
      Net income                                            $  526           $  610 
                                                            ------           ------ 
      Weighted average shares                                4,749            4,562 
                                                            ------           ------ 
      Basic earnings per share                              $  .11           $  .13 
                                                            ------           ------ 
                                                                                    
      DILUTED                                                                       
      Net income                                            $  526           $  610 
                                                            ------           ------ 
      Weighted average shares                                4,749            4,562 
      Effect of:                                                                    
              Stock options                                    362              247 
                                                            ------           ------ 
      Weighted average shares, as adjusted                   5,111            4,809 
                                                            ------           ------ 
                                                                                    
      Diluted earnings per share                            $  .10           $  .13 
                                                            ======           ====== 
</TABLE>
                                                                              

4.    INCOME TAXES
      At March 31, 1998, the Company had approximately $3,286,000 of federal net
      operating loss carryforwards. The carryforwards expire through the year
      2010. The Company also has unused investment tax and other credits of
      approximately $107,000 expiring through 2001. The effective tax rate of
      40% exceeded the statutory federal income tax rate of 34% primarily due to
      the impact of state income taxes.





                                      -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 first quarter of fiscal 1999 net sales and contract revenues grew to
$11,692,000, a 55% increase from the comparable period a year ago and a 37%
increase from the previous quarter. The company earned net profits of $526,000
in the current quarter, compared to a net profit of $610,000 in the
corresponding period a year ago and a net profit of $2,615,000 in the previous
quarter. This decline in profitability is due to an increase in the effective
tax rate to 40% in the current quarter from approximately 4% in fiscal 1998, as
well as the one-time effect of a $2,072,000 tax benefit in the fourth quarter of
fiscal 1998 associated with recognition of net operating loss carryforwards and
other tax assets.


RESULTS OF OPERATIONS
Net sales and contract revenues in the first quarter increased by $4,160,000
(55%) in comparison to the corresponding year ago period and $3,130,000 (37%)
compared to the fourth quarter of fiscal 1998. Compared to the previous quarter,
security systems and related field service revenues increased by $2,687,000
(34%) and research and engineering revenues improved by $443,000 (71%). The
increase in security equipment and related field service revenues is primarily
due to increased sales of X-ray equipment.

For the first quarter, costs of sales and contracts increased to $7,239,000 from
$4,782,000 in the corresponding period a year ago due primarily to increased
sales volume. Costs of sales and contracts represented 62% of revenues versus
63% for the corresponding period last year and 60% for the fourth quarter of
fiscal year 1998. The costs of sales percentage of revenues in the current
quarter increased from the previous quarter primarily due to sales mix.

Selling, general and administrative expenses of $2,131,000 for the first quarter
were higher by 28% compared to the corresponding year-ago period and higher by
12% compared to the fourth quarter of fiscal 1998. As a percent of sales,
selling, general and administrative expenses were 18% of revenues in the current
quarter compared to 22% of revenues for the corresponding year-ago period and
22% for the fourth quarter of fiscal year 1998. The decrease is due primarily to
an increased sales base over which overhead costs are spread.

Company-funded research and development expenses of $1,408,000 for the first
quarter were higher by $944,000 (203%) compared to the year-ago quarter and
higher by $466,000 (49%) compared to the fourth quarter of fiscal year 1998. The
increase is an intentional result of the Company's commitment to research and
development.

The Company produced a net profit of $526,000 during the first quarter. This is
a decline of $84,000 (14%) over net profit in the year-ago quarter and
$2,089,000 (80%) from the fourth quarter of fiscal 1998. The decline in
profitability is the result of a one-time tax benefit of $2,072,000 associated
with net operating loss carryforwards and other tax assets recorded in the
fourth quarter of fiscal 1998 as well as a change in the effective tax rate from
approximately 4% in fiscal 1998 to 40% in fiscal 1999.






                                      -8-
<PAGE>   9

LIQUIDITY AND CAPITAL RESOURCES

Cash and cash equivalents decreased by $889,000 to $1,401,000 at June 30, 1998
compared to $2,290,000 on March 31, 1998. This decrease was primarily due to an
increase in unbilled costs and fees and accounts receivable of $1,765,000 and
$805,000, respectively. Working capital increased by $425,000 (3%) since March
31, 1998, growing from $14,691,000 to $15,116,000 at the end of the first
quarter.

During April 1998, the Company renegotiated the line of credit, from $8.0
million to $12.0 million in anticipation of the increased standby letter of
credit capacity required to support the growth in international orders. The
Company's credit facility restricts the payment of dividends, except in shares
of the Company's stock, without the consent of the bank.

Given the Company's receipt of various international contract awards totaling
$45 million, management believes that additional credit facilities will be
necessary for expanded standby letter of credit capacity to support foreign
equipment sales. Management is currently in the process of negotiating with two
commercial banks and the U.S. Export Import Bank to increase its bank lines of
credit and anticipates having a new credit facility in place during the second
quarter of fiscal year 1999.

At June 30, 1998, the company had external borrowings, with a local bank, in the
amount of $1,000,000. Interest is payable monthly at the bank's prime rate of
interest.




                                      -9-
<PAGE>   10


                     AMERICAN SCIENCE AND ENGINEERING, INC.


Part II - Other Information

Item 6.  EXHIBITS AND REPORTS ON FORM 8-K

(a)      REPORTS ON FORM 8-K

         A report on Form 8-K was filed on April 17, 1998 concerning the
         Company's adoption of a Shareholders Rights Plan.


                                   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)


                                   /s/ Lee C. Steele
Date: 11 August 1998               ------------------------------------------
                                   Lee C. Steele
                                   Vice President and Chief Financial Officer






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