Model checking and refinement checking are two approaches to formal verification, whose difficulties are due to the state explosion problem. As one of the proposed solutions to the problem, it is suggested to introduce compositionality in model checking and refinement checking based on the idea of divide-and-conquer, by which the verification task of the whole system is decomposed to several smaller subtasks on the subsystems. In a uniform framework, this paper surveys the approaches of compositional model checking and compositional refinement checking in a systematic way. From the perspective of module checking, the principle and verification strategies of the two compositional verification approaches are introduced. In addition, the complexities of various kinds of related problems are summarized and a comparison is made between compositional model checking and compositional refinement checking, which exposes the intrinsic relation between them. Finally, some trends are given for the future research.