Let p be a prime number. Roughly speaking, rigid analytic geometry is a counterpart of complex analysis where one replaces the field C of complex numbers by the field Qp of p-adic rational numbers (or some extension thereof).   In this talk, I'll try to explain some of the motivations which led to the development of this theory, and to give some flavor of the recent progress which make it a timely subject for our Special Year. No prior knowledge of p-adic geometry will be assumed.