**EXHIBIT 31.1** **CERTIFICATION**     I, Kevin T. Conroy, President and Chief Executive Officer of    Third Wave Technologies, Inc. (the “registrant”),    certify that:     1. I have reviewed this Annual Report on    Form 10-K    of the registrant;     2. Based on my knowledge, this annual 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))    and internal control over financial reporting [as defined in    Exchange Act    Rules 13a-15(f)    and    15d-15(f)    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. designed such internal control over financial reporting,    or caused such internal control over financial reporting to be    designed under our supervision, to provide reasonable assurance    regarding the reliability of financial reporting and the    preparation of financial statements for external purposes in    accordance with generally accepted accounting principles;     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 most recent fiscal quarter    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 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 data; 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. /s/   Kevin T. ConroyKevin T. Conroy Date: March 9,2006