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                                                                    Exhibit 31.1

                                  CERTIFICATION

I, Thomas P. Richards, certify that:

1.    I have reviewed this quarterly report of Form 10-Q of Grey Wolf, Inc;

2.    Based on my knowledge, this report does not contain any untrue statement
      of a material fact or omit to state a material fact necessary to make the
      statements made, in light of the circumstances under which such statements
      were made, not misleading with respect to the period covered by this
      report;

3.    Based on my knowledge, the financial statements, and other financial
      information included in this report, fairly present in all material
      respects the financial condition, results of operations and cash flows of
      the registrant as of, and for, the periods presented in this report;

4.    The registrant's other certifying officer and I are responsible for
      establishing and maintaining disclosure controls and procedures (as
      defined in Exchange Act Rules 13a-15(e) and 15d-15(e)) for the registrant
      and have:

      (a)   designed such disclosure controls and procedures, or caused such
            disclosure controls and procedures to be designed under our
            supervision, to ensure that material information relating to the
            registrant, including its consolidated subsidiaries, is made known
            to us by others within those entities, particularly during the
            period in which this report is being prepared;

      (b)   [paragraph omitted pursuant to SEC Release Nos. 33-8238 and
            34-47986];

      (c)   evaluated the effectiveness of the registrant's disclosure controls
            and procedures and presented in this report our conclusions about
            the effectiveness of the disclosure controls and procedures, as of
            the end of the period covered by this report based on such
            evaluation; and

      (d)   disclosed in this report any change in the registrant's internal
            control over financial reporting that occurred during the
            registrant's quarter ended March 31, 2004 that has materially
            affected, or is reasonably likely to materially affect, the
            registrant's internal control over financial reporting; and

5.    The registrant's other certifying officer and I have disclosed, based on
      our most recent evaluation of internal control over financial reporting,
      to the registrant's auditors and the audit committee of the registrant's
      board of directors (or persons performing the equivalent functions):

      (a)   all significant deficiencies and material weaknesses in the design
            or operation of internal control over financial reporting which are
            reasonably likely to adversely affect the registrant's ability to
            record, process, summarize and report financial information; and

      (b)   any fraud, whether or not material, that involves management or
            other employees who have a significant role in the registrant's
            internal control over financial reporting.

Date: May 5, 2004                      By: /s/ Thomas P. Richards
                                           -------------------------------------
                                           Thomas P. Richards
                                           Chairman, President and Chief
                                           Executive Officer