This chapter presents POPMUSIC, a general decomposition-based framework within the realm of matheuristics that has been successfully applied to various combinatorial optimization problems. POPMUSIC is especially useful for designing heuristic methods for large combinatorial problems that can be partially optimized. The basic idea is to optimize subparts of solutions until a local optimum is reached. Implementations of the technique to various problems show its broad applicability and efficiency for tackling especially large-size instances.