This book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with emphasis on applications in mathematical physics (especially, Schrodinger operators) and analysis (Dirichlet and Neumann LaPlacians, Sturm-Liouville operators, Hamburger moment problems). Among others a number of advanced special topics are treated on a text book level accompanied by numerous illustrating examples and exercises. The main themes of the book are the following: -Spectral integrals and spectral decompositions of self-adjoint and normal operators -Perturbations of self-adjointness and spectra of self-adjoint operators -Forms and operators -Self-adjoint extension theory: boundary triplets, Krein-Birman-Vishik theory of positive self-adjoint extensions.