Two of the most central algorithmic paradigms implemented in practical solvers for maximum satisfiability (MaxSAT) and other related declarative paradigms for NP-hard combinatorial optimization are the core-guided (CG) and implicit hitting set (IHS) approaches. We develop a general unifying algorithmic framework, based on the recent notion of abstract cores, that captures both CG and IHS computations. The framework offers a unified way of establishing the correctness of variants of the approaches, and can be instantiated in novel ways giving rise to new algorithmic variants of the core-guided and IHS approaches. We illustrate the latter aspect by developing a prototype implementation of an algorithm variant for MaxSAT based on the framework.