A theorem of Ivanov states that the mapping class group of a finite-type surface is also the automorphism group of a simplicial complex associated to the surface, the complex of curves. In other words, any automorphism of the complex of curves is somewhat "rigid", since it can only come from a homeomorphism of the surface. This fact, which is the starting point of the geometric group theory of mapping class groups, can then be used to prove other "rigidity" results, such as that every quasi-isometry is within finite Hausdorff distance from the multiplication by some group element, and that every group automorphism is inner. In this talk, we first review the literature on the above results, giving a sketch of how one can see them as "corollaries" of Ivanov's theorem. Then we show that, assuming a forthcoming result of Abbott-Berlyne-Ng-Rasmussen, the same type of properties are enjoyed by random quotients of mapping class groups.