This book provides a concise introduction to the mathematical cosmology of the origin, structure and evolution of the universe. The book begins with a brief overview of observational and theoretical cosmology, along with a short introduction to general relativity. It then goes on to discuss Friedmann models, the Hubble constant and deceleration parametr, singularities, the early universe, inflation, quantum cosmology and the disatant future of the universe. This new edition contains a rigorous derivation of the Robertson-Walker metric. It also discusses the limits to the parametr space through various theoretical and observational constrains, and presents a new inflationary solution for a sixth degree potential.