Bestseller of the XXth Century in Mathematical Physics voted on by participants of the XIIIth International Congress on Mathematical Physics. This revision will make this book more attractive as a textbook in functional analysis. Further refinement of coverage of physical topics will also reinforce its well-established use as a course book in mathematical physics. This book covers the theory of eigenvalues of Schrodinger operators. It clearly explains the basic concepts involved: perturbation theory (summability questions, Fermi's golden rule), min-max principle for discrete spectrum, Weyl theorem, HVZ theorem, the absence of singular continuous spectrum, ground state questions, periodic operators, semi-classic distribution of eigenvalues, compactness criteria.