Rigid Ricci solitons have been recently introduced by Petersen and Wylie. It is a remarkable fact that under certain geometric conditions (like homogeneity or locally conformal flatness) any gradient shrinking Ricci soliton is necessarily rigid. The aim of this lecture is to present a characterization of rigid shrinking Ricci solitons in terms of the harmonicity of the Weyl tensor as well as some sufficient conditions for rigidity of gradient Ricci solitons. Special attention will be paid to the differences between the compact and complete noncompact cases.