The course is an introduction to the theory of unbounded operators in Hilbert spaces and consists of two parts: Part 1 develops the general theory of unbounded operators. The main topics here are domains, graphs, adjoint operators, spectrum, resolvent, symmetric operators and quadratic forms, symmetric extensions, deficiency indices, self-adjoint operators, Cayley transform, Spectral theorem, Stone theorem. Part 2 is an introduction to the spectral theory of differential operators (Sturm-Liouville operators and Hill-Schrödinger operators). The main topics include domains, spectra localization, asymptotics of eigenvalues and eigenfunctions, bases of root functions, convergence of spectral decompositions.