Résume | The space of finitely generated marked groups, denoted by G, is a locally compact Polish space whose elements are groups with fixed finite generating sets; the topology on G is induced by the local convergence of the corresponding Caley graphs. We will discuss equivalent characterizations of closed subspaces S of G satisfying the following zero-one law: for any sentence sigma in the infinitary logic L_{\omega_1, \omega}, the set of all models of sigma in S is either meager or comeager. In particular, this zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. We will also discuss some open problems. |