This course aims to introduce students to classical and quantum mechanics by means of a mathematically rigorous approach. In the first part of the course we shortly introduce Newtonian mechanics and develop Lagrangian and Hamiltonian mechanics on manifolds with a particular emphasis on the underlying variational principles.  The second part introduces Quantum Mechanics in an axiomatic way and is focused on the study of  the spectral properties of the Schrödinger Hamiltonian for different classes of potentials.