Let a topological group G act continuously on a topological space X. The action is "locally faithful" if the intersection of the stabilizers is discrete in G. The action is " orbit nonproper" if, for some point x in X, the orbit map g ® gx : G ® X is nonproper. If X is compact, then any continuous action on X is nonproper. Any action with an orbit that is not closed is necessarily an orbit nonproper action. Thus " orbit nonproper" is a very mild condition that includes all but the simplest kinds of dynamics. In this talk, I will describe the collection of connected, simply connected Lie groups that admit a locally faithful, orbit nonproper action by isometries of a connected Lorentz manifold. The flavor of the result is that even such tame dynamical conditions impose serious restrictions on the list of groups. I will also describe similar work of:

* R. Zimmer for semisimple groups on compact connected Lorentz manifolds,

* M. Gromov for nilpotent groups on compact connected Lorentz manifolds,

* N. Kowalsky for simple groups on connected Lorentz manifolds.