Understanding quantum mechanics inevitably leads to an in-depth study of the Schr?dinger operator. This set of review lectures informs researchers and advanced students of the most recent developments in the analysis of the Schr?dinger operator occurring in solid-state physics, nuclear physics, etc. The topics covered are nonlinear and random potentials, magnetic fields, and many-body problems. Inverse spectral theory is also treated. The results are mathematically rigorous and many physical implications are discussed. The book is suitable for advanced courses in mathematical physics.