Given two infinite groups G_1,G_2 and two finite presentations describing them, how can we tell if G_1 and G_2 are isomorphic? Given a single group G, how can we describe its outer automorphism group Out(G)? These two fundamental problems are closely related, and were shown to be undecidable in complete generality. At the same time, for families of groups naturally arising in topology and geometry, a solution is sometimes possible. I will discuss what is known on these problems, with a special emphasis on negatively-curved and non-positively curved groups.