Stein domains in ℂ2 with prescribed boundary

Abstract We give an overview of the research related to the topological characterization of Stein domains in complex two-dimensional space, and an instance of their many important connections to smooth manifold topology in dimension four. One goal is to motivate and explain the following remarkable conjecture of Gompf: no Brieskorn integral homology sphere (other than S3) admits a pseudoconvex embedding in ℂ2, with either orientation. We include some new examples and results that consider the conjecture for families of rational homology spheres which are Seifert fibered, and integral homology spheres which are hyperbolic.