In this introductory chapter we describe in detail the quantum mechanical concepts and relations to be used for deriving the basic principles of statistical mechanics as well as its advanced applications to realistic molecular systems. After introducing the time-dependent Schroedinger equationSchroedinger equation, its general analytical solution for stationary HamiltoniansStationary Hamiltonian and the Born–Oppenheimer approximationBorn–Oppenheimer approximation, we specifically address the treatment of molecular systems characterized by the coexistence of quantum and classical-like degrees of freedom (mixed quantum-classical systemsMixed quantum-classical systems).