This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. Its central aim is to explain Thurston's geometrisation theorem for three-manifolds, proved by Perelman in 2002. The text is divided into three parts: hyperbolic geometry, surfaces, and three-manifolds. It includes complete proofs of Mostow rigidity, the thick-thin decomposition, Thurston's classification of surface diffeomorphisms via Bonahon's theory of geodesic currents, the prime and JSJ decompositions, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling theorem. Suitable for postgraduate students in mathematics interested in geometry and topology, this book will also be valuable to mathematicians seeking a rigorous introduction to the field. Readers are assumed to have a standard background in topology and analysis.