Transformation groups as symmetries of manifolds link geometry and groups and have found multiple applications in the study of Riemmanian manifolds. In this talk I will discuss topological and geometric applications of transformation groups in the more general setting of Alexandrov spaces (with curvature bounded below). These spaces are a synthetic generalization of Riemannian manifolds with a lower curvature bound and arise naturally, for example, as orbit spaces of isometric group actions on Riemannian manifolds with curvature bounded below.