The authors prove striking results on upper semicontinuity of automorphism groups. They show, for example, that if pairs \((M_j, G_j)\) \((M_j\) connected complex manifolds and \(G_j\) subgroups of \(\Aut (M_j))\) converge on compacta to a pair \((M,G)\), where \(M\) is a hyperbolic complex manifold and \(G\) is a subgroup of \(\Aut (M)\), then \(G_j\) is isomorphic to \(G\) for large \(j\). They also show that any domain \(M\) in \(\mathbb{C}^n\) can be represented as a union of an increasing sequence of bounded domains \(M_k\) which admit \(\mathbb{Z}_k\) as automorphisms.