FORM 6-K


                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D. C. 20549



                        Report of Foreign Private Issuer
                      Pursuant to Rule 13a-16 or 15d-16 of
                       the Securities Exchange Act of 1934

                           For the month of April 2004

                           A/S STEAMSHIP COMPANY TORM
                 (Translation of registrant's name into English)

                                   Marina Park
                                Sundkrogsgade 10
                              DK-2100 Copenhagen 0
                                     Denmark
                    (Address of principal executive offices)

     Indicate by check mark whether the registrant files or will file annual
reports under cover Form 20-F or Form 40-F.

                             Form 20-F  X       Form 40-F
                                      -----              -----

     Indicate by check mark whether the registrant by furnishing the information
contained in this Form is also thereby furnishing the information to the
commission pursuant to Rule 12g3-2(b) under the Securities Exchange Act of 1934.

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



INFORMATION CONTAINED IN THIS FORM 6-K REPORT

     Set forth herein 1) as Exhibit 1 is a copy of Share Statement No. 01 - 2004
issued by A/S STEAMSHIP COMPANY TORM (the "Company") to The Copenhagen Stock
Exchange (the "Exchange") on March 16, 2004, 2) as Exhibit 2 is a copy of Share
Statement No. 04 - 2004 issued by the Company to the Exchange on March 29, 2004,
3) as Exhibit 3 is a copy of Share Statement No. 05 - 2004 issued by the Company
to the Exchange on April 1, 2004 and 4) as Exhibit 4 is a copy of Share
Statement No. 06-2004 issued by the Company to the Exchange on April 2, 2004.





Exhibit 1




SHARE STATEMENT No. 1 -2004

16 March 2004


This is to notify you of changes in the shareholdings of the company's insiders
pursuant according to the Danish Securities Trading Act.

                                                    Total       Total
Securities     Trade        Holding   Market        Holding     market
code           date         number    value         number      value
----           ----         ------    -----         ------      -----
DK0010281468   12.03.2004   -1,000    350,000.00    1,162,299   446,148,352.00


A/S Dampskibsselskabet TORM
Contact person:   Klaus Nyborg, CFO (tel.: +45 39 17 92 00)





Exhibit 2




SHARE STATEMENT No. 4  -2004

29 March 2004


This is to notify you of changes in the shareholdings of the company's insiders
pursuant according to the Danish Securities Trading Act.


                                                    Total       Total
Securities     Trade        Holding   Market        Holding     market
code           date         number    value         number      value
----           ----         ------    -----         ------      -----
DK0010281468   26.03.2004   +12,348   4,344,661.60  1,224,122   437,819,392.00



A/S Dampskibsselskabet TORM
Contact person:   Klaus Nyborg, CFO (tel.: +45 39 17 92 00)





Exhibit 3




SHARE STATEMENT No. 5  -2004

1 April 2004


This is to notify you of changes in the shareholdings of the company's insiders
pursuant according to the Danish Securities Trading Act.

                                                    Total       Total
Securities     Trade        Holding   Market        Holding     market
code           date         number    value         number      value
----           ----         ------    -----         ------      -----
DK0010281468   31.03.2004   -6,400    2,274,575.00  1,217,722   433,959,424.00




A/S Dampskibsselskabet TORM
Contact person:   Klaus Nyborg, CFO (tel.: +45 39 17 92 00)





Exhibit 4




SHARE STATEMENT No. 6  -2004

2 April 2004


This is to notify you of changes in the shareholdings of the company's insiders
pursuant according to the Danish Securities Trading Act.


                                                    Total       Total
Securities     Trade        Holding   Market        Holding     market
code           date         number    value         number      value
----           ----         ------    -----         ------      -----
DK0010281468   01.04.2004   -16,000   5,600,000.00  1,201,722   419,857,408.00



A/S Dampskibsselskabet TORM
Contact person:   Klaus Nyborg, CFO (tel.: +45 39 17 92 00)





                                   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, thereunto duly authorized.

                           A/S STEAMSHIP COMPANY TORM
                                  (registrant)



Dated: April 2, 2004                                By: /s/ Klaus Nyborg
                                                       ------------------
                                                            Klaus Nyborg
                                                       Chief Financial Officer



03810.0001 #476148