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 June, 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 a Share Statement No. 11
- 2004 issued by A/S STEAMSHIP COMPANY TORM (the "Company") to The Copenhagen
Stock Exchange (the "Exchange") on June 2, 2004, 2) as Exhibit 2 is a copy of a
Share Statement No. 12 - 2004 issued by the Company to the Exchange on June 18,
2004.


Exhibit 1

SHARE STATEMENT No. 11 - 2004

02 June 2004

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



-----------------------------------------------------------------------------------------------
Securities code     Trade date  Holding       Market        Total Holding      Total market
                                 number       value             number             value
-----------------------------------------------------------------------------------------------
                                                                
  DK0010281468      01.06.2004  -3,500      524,475.00        2,369,966        349,000,960.00
-----------------------------------------------------------------------------------------------


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


Exhibit 2

SHARE STATEMENT No. 12-2004

18 June 2004

Please below find the total shareholdings of all insiders and related parties,
including the shareholding of the entire Board of Directors (including related
parties) and the entire Executive Board of Directors (including related parties)
pursuant to the Danish Securities Trading Act.

--------------------------------------------------------------------------------
           Securities code            Holding                   Market value
            DK0010281468              (number)                      (DKK)
                                 As at 16 June 2004          As at 16 June 2004
--------------------------------------------------------------------------------
Board of Directors                         18,098                  2,618,418.64
(incl. related parties)
--------------------------------------------------------------------------------
Executive Board of Directors                9,296                  1,344,945.28
(incl. related parties)
--------------------------------------------------------------------------------
All (incl. related parties)             2,361,766                341,700,304.88
--------------------------------------------------------------------------------

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: June 24, 2004                             By: /s/ Klaus Nyborg
                                                     --------------------------
                                                 Klaus Nyborg
                                                 Chief Financial Officer

03810.0001 #495381