Emerging as a relatively new class of porous materials, metal-organic frameworks (MOFs), possessing diversified, designable and tailorable structures as well as ultrahigh surface area, have captured broad research interest and shown potential applications in many fields in recent years. In particular, MOFs have attracted intensive attention in catalysis. In the first two parts of this review, according to the origin of active sites, for examples, coordinatively unsaturated metal centers, functional organic linkers, functional sites chemically grafted onto the framework, as well as metal complexes or metal nanoparticles (MNPs) encapsulated inside the MOFs, etc., we have summarized the recent progress in heterogeneous catalysis over MOFs and their composites in recent several years. In addition, the MOF-based photocatalysis and electrocatalysis have also been briefly introduced in the subsequent two parts. Finally, the further development and challenge in MOF catalysis are discussed.