This two-volume work established itself as a standard reference serving both as a self-contained, advanced textbook, and as a research tool. Volume 1 addresses fundamental models of non-relativistic and relativistic quantum mechanics including electrostatic potentials, magnetic fields and spin, offering a systematic, rigorous description of both classical and recent developments, which can be a point of departure for further research. It combines and further develops cutting edge mathematical tools, involving functional analysis, stochastic processes with continuous paths or jumps, and stochastic analysis, while the presentation is user-friendly. A new section is now added on the description of bound states at short to mid-range complementing the asymptotic analysis at infinity, thus providing a full picture on the spatial behaviour of eigenfunctions.