Given a discrete group G and a family F of subgroups of G there is a G-CW complex that classifies G-CW complexes with isotropy contained in the family F. Such space is unique up to G-equivariant homotopy and is often called the classifying space of G for the family F. For the trivial family it is the universal cover of a K(G,1) space, and more generally, classifying spaces for families play an important role in the classification of manifolds with a given fundamental group G. In this talk, we will introduce these notions and survey what is known about classifying spaces for some families of subgroups of the mapping class group of an orientable surface of finite type. Time permitting, we will discuss recent joint work with Porfirio León Álvarez and Luis Jorge Sánchez Saldaña in this topic.