The third edition of this textbook is fully updated and includes important developments in cosmology. It begins with an introduction to cosmology and general relativity, and goes on to cover the mathematical models of standard cosmology. The physical aspects of cosmology, including primordial nucleosynthesis, the astroparticle physics of inflation, and the current ideas on structure formation are discussed. Alternative models of cosmology are reviewed, including the model of Quasi-Steady State Cosmology, which has been proposed as an alternative to Big Bang Cosmology. The final chapters discuss observational cosmology, and debate theory versus observational methods. The book includes 400 problems and numerous worked examples. This introductory textbook describes modern cosmology at a level suitable for advanced undergraduates and graduates who are familiar with mathematical methods and basic theoretical physics. It is intended for use on courses in theoretical physics, astrophysics or applied mathematics that include modern cosmology.