Abstract: Rigidity theory of circle diffeomorphisms, which concerns smooth conjugacy to a rigid rotation, is a classic problem in dynamical systems initiated by Arnol'd and settled by Herman and Yoccoz. We present complete renormalization and rigidity theory for circle maps with breaks, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity. We prove that renormalizations of any two C^{2+alpha}-smooth (alpha>0) circle maps with breaks, with the same irrational rotation number and the same size of the break, approach each other exponentially fast. As a corollary, we obtain a strong rigidity statement for such maps: for almost all irrational rotation numbers, any two circle maps with breaks, with the same rotation number and the same size of the break, are C^1-smoothly conjugate to each other. As we proved earlier, the latter result cannot be extended to all irrational rotation numbers. (This joint work with Kostya Khanin)