These notes have been finished at INRIA during my visit of the project ONDES in January-February 2003. They correspond to a one-term course given at the University of Vienna in 2002 year. The aim of the notes is to give an introduction to Classical Field Theory and Quantum Mechanics for mathematicians. We start with the Lagrangian and Hamiltonian Theory of finite-dimensional systems and general nonlinear wave equations. Next we prove the Noether Theory of Invariants for the Fields and expose the Lagrangian form of the Maxwell Electrodynamics. Then we give a concise introduction of the Schrödinger Equation for the electron in the Maxwell Field: the equation describes the particle-like properties of cathode rays through the short-wave WKB-asymptotics. Finally, we apply the Schrödinger Equation to the derivation of spectrum of Hydrogen atom, dipole radiation and selection rules in a uniform magnetic field.