We develop a novel framework, the implicit hitting set approach, for solving a class of combinatorial optimization problems. The explicit hitting set problem is as follows: given a set U and a family S of subsets of U, find a minimum-cardinality set that intersects (hits) every set in S. In the implicit hitting set problem (IHSP), the family of subsets S is not explicitly listed (its size is, generally, exponential in terms of the size of U); instead, it is given via a polynomial-time oracle that verifies if a given set H is a hitting set or returns a set in S that is not hit by H. Many NP-hard problems can be straightforwardly formulated as implicit hitting set problems. We show that the implicit hitting set approach is valuable in developing exact and heuristic algorithms for solving this class of combinatorial optimization problems. Specifically, we provide a generic algorithmic strategy, which combines efficient heuristics and exact methods, to solve any IHSP. Given an instance of an IHSP, the proposed algorithmic strategy gives a sequence of feasible solutions and lower bounds on the optimal solution value and ultimately yields an optimal solution. We specialize this algorithmic strategy to solve the multigenome alignment problem and present computational results that illustrate the effectiveness of the implicit hitting set approach.