A classical result of Myers and Steenrod states that the isometry group of a compact Riemannian manifold is a compact Lie transformation group. Also classical is the fact that this compactness property fails for general pseudo-Riemannian manifolds, allowing interesting dynamics for the group of isometries. In this talk, we will be interested by the topological, and dynamical consequences of the noncompactness of the isometry group. We will especially focus on the case of Lorentz manifolds, and we will present a complete topological classification of 3-dimensional closed Lorentz manifolds having a noncompact isometry group.