Let \(D_ 0\) be a \(C^{\infty}\) strongly pseudoconvex bounded domain in \({\mathbb{C}}^ n\) (or in a Stein manifold). The authors study the holomorphic automorphism groups Aut(D) of like domains that are \(C^{\infty}\) close to \(D_ 0\). Their main results are as follows: If D is sufficiently \(C^{\infty}\) close to \(D_ 0\), then Aut(D) is isomorphic to a subgroup of \(Aut(D_ 0)\), with an isomorphism given by \(\alpha \to f{\mathbb{O}}\alpha {\mathbb{O}}f^{-1}\) where f is a \(C^{\infty}\) diffeomorphism of D onto \(D_ 0\). If \(D_ 0\) is not biholomorphic to the ball, then \(C^{\infty}\) arbitrarily close to \(D_ 0\) there is a domain \(D\subset D_ 0\) with real analytic boundary such that Aut(D) consists of the restrictions of the members of \(Aut(D_ 0)\) to D; in particular, these two automorphism groups are isomorphic. This paper begins with an excellent exposition of the intuitive meaning and consequences of these results. A precise definition of the \(C^{\infty}\) topology used here was given by the authors in an earlier paper [Adv. Math. 43, 1-86 (1982; Zbl 0504.32016)].