This paper defines coverT(V,F) (also called the cover of formula F with respect to a set of variables V in theory T to be the strongest quantifier-free formula that is implied by \exists(V): F in theory T. Cover exists for several useful theories, including those that do not admit quantifier elimination. This paper describes cover algorithms for the theories of uninterpreted functions and linear arithmetic. In addition, the paper provides a combination algorithm to combine the cover operations for theories that satisfy some general condition. This paper motivates the study of cover by describing its applications in program analysis and verification techniques, like symbolic model checking and abstract interpretation.