  This paper collects and extends the lectures I gave at the "XXIV
International Fall Workshop on Geometry and Physics" held in Zaragoza (Spain)
August 31 - September 4, 2015. Within these lectures I review the formulation
of Quantum Mechanics, and quantum theories in general, from a mathematically
advanced viewpoint, essentially based on the orthomodular lattice of elementary
propositions, discussing some fundamental ideas, mathematical tools and
theorems also related to the representation of physical symmetries. The final
step consists of an elementary introduction the so-called (C*-) algebraic
formulation of quantum theories.
