Stein manifolds are affine complex manifolds; their study has been a classical subject of complex analysis for the past century. The field took a surprising turn in the 1990ies when Y. Eliashberg used symplectic methods to achieve a topological characterization of Stein manifolds. Recent progress in symplectic topology has led to the discovery of exotic Stein structures on complex linear space. In the converse direction, work of S. Donaldson has exhibited Stein manifolds as the central building blocks for closed symplectic manifolds.