This book introduces aspects of topology and their applications to problems in condensed matter physics. Basic topics in mathematics have been introduced in a form accessible to physicists, and the use of topology in quantum, statistical and solid state physics has been developed with an emphasis on pedagogy. The aim is to bridge the language barrier between physics and mathematics, as well as the different specializations in physics. The book consists roughly of two parts: one corresponds to developing the necessary mathematics and the other discusses applications to physical problems. The section on mathematics is a quick, but more-or-less complete, review of topology. The focus is on explaining fundamental concepts rather than dwelling on details of proofs, while retaining a mathematical flavour. The physics section starts with an introduction and goes on to discuss topics from quantum mechanics, statistical mechanics of polymers, knots and vertex models, solid state physics, and exotic excitations such as Dirac quasiparticles, Majorana modes, Abelian and non-Abelian anyons. Quantum spin liquids and quantum information-processing have also been treated in some detail. Pitched at the level of a graduate student of physics, this book does not assume any additional knowledge of mathematics or physics. It is therefore suitable for advanced Masters' students as well. A collection of selected problems will help the reader learn the topics on ones own, and the broad range of topics covered will make the text a valuable resource for practicing researchers in the field.