Ricci solitons are Riemannian metrics satisfying a certain equation (Ric + (1/2)L_X(g) = λg) that evolve under the Ricci flow by diffeomorphisms and scaling. They are a generalization of Einstein metrics (Ric = λg, evolve under RF by scaling). Symmetry assumptions often make analysis easier, and it seems natural to study Ricci soliton metrics that have lots of isometries. However, having lots of isometries is a strong condition on a soliton metric. In this talk I will present a theorem of Petersen and Wylie which says that a homogeneous gradient soliton metric must be rigid, i.e. isometric to (a quotient of) the product of an Einstein manifold with a Euclidean factor.