**Exhibit 31.1** **CERTIFICATION    BY CHIEF EXECUTIVE OFFICER**     I, Frank J. Bramanti, certify that:     1.     I have reviewed this annual report on    Form 10-K    of HCC Insurance Holdings, 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 in order 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 (the    registrant’s fourth fiscal quarter in the case of an annual    report) 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 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.        February 29, 2008 /s/  Frank    J. Bramanti     Date     Frank J. Bramanti, Chief Executive Officer