The rigidity problem for homeomorphisms is one of the most classical questions of low-dimensional dynamics. It asks, under what conditions, the combinatorial equivalence of two maps translates into a smooth conjugacy between them. Of particular interest is the case of smooth homeomorphisms with finitely many critical points (multicritical circle maps), because of its connection with Renormalization Theory. I will review the current state of the art and will present my recent work with I. Gorbovickis, which settles a major part of the rigidity problem for multicritical circle maps.