By classical results of Rochlin, Thom, Wallace and Lickorish, it is well-known that any two 3-manifolds (with diffeomorphic boundaries) are related one to the other by surgeries. Yet, by restricting the types of the surgeries, one can define several families of non-trivial equivalence relations on the sets of (diffeomorphism classes of) 3-manifolds. In this expository paper, which is based on lectures given at the school "Winter Braids XI" (Dijon, December 2021), we explain how certain filtrations of mapping class groups of surfaces enter into the definitions and the mutual comparison of these surgery equivalence relations. We also survey the ways how concrete invariants of 3-manifolds (such as finite-type invariants) can be used to characterize such relations.