In this talk, I will introduce Stein manifolds from the point of view of symplectic topology. In a sense these provide a unified framework for studying both affine algebraic geometry and smooth manifold theory. While the existence of Stein manifolds with given topology has been understood since the nineties, the uniqueness problem is highly intricate and still poorly understood. I will describe what is known and survey some recent progress. Our main result is a construction which produces exotic Stein manifolds and also an invariant for distinguishing them. This is partly based on joint work with Emmy Murphy.