Kapranov has proposed the following informal principle: begin with a variety X(0), and let X(1) be the moduli space of deformations of X(0), X(2) the moduli space of deformations of X(1), and so on. Then this process should stop after d = dim X steps, i.e., X(d) should be rigid (no deformations). We prove a precise formulation of this principle in the case d=1: we show that the moduli stack of stable curves of genus g with n marked points is rigid for each g and n. We also describe some ideas and examples in the case d=2.