The aim of this course is to give an introduction to smooth 4-manifolds. We will study the geometry of some interesting complex surfaces as well, including elliptic surfaces, genus 2 fibrations, fake complex projective planes and the Cartwright-Steger surface. Topics will also include Lefshetz fibrations, symplectic 4-manifolds, Seiberg-Witten invariants, the rational blowdown operation and exotic smooth structures on 4-manifolds.