This is a textbook on non-relativistic quantum mechanics that emphasizes clarification of the nature of the basic postulates and the interpretation of the theory. It contains special material, often only accessible in scientific journals, on bound states, scattering theory, and both analytical and approximation techniques. Applications to many branches of physics are given. The book is a considerably improved and completely updated English translation of a very successful Spanish textbook and is aimed at students in their second year of university.