As is well-known, the theory of Quantum Mechanics, shortly after its birth in the years 1925-26, has undergone an important process of mathematical formalization, mainly due to the work of Von Neumann, culminated with his famous treatise of 1932. This formalization did not consist in the application of existing mathematical theories, but was accompanied by the development of the required theories from functional analysis. The mathematical language of the theory of unbounded operators on Hilbert spaces is much more abstract and difficult than those used by many classical physical theories. Therefore, in many presentations of quantum mechanics, this language is often avoided, as far as possible. However, the unavoidable use of at least a portion of the mathematical language related to the formalization of quantum mechanics can leave to the student an insatisfactory sense of mistery. These lessons aim to introduce the student to the understanding of the mathematical formulation of the axioms and the mathematical treatment of some fundamental concepts of quantum mechanics. The hope is that the logical coherence, unity and nonarbitrariness of this mathematical theory may be appreciated. In order to achieve this goal, it is necessary to go beyond the very basic theory of Hilbert spaces given in the course of "Mathematical methods of Engeneering", introducing some concepts of the theory of unbounded operators on Hilbert spaces. After that, we will be able discuss the mathematical axioms of quantum mechanics and revise some basic concepts which the student already knows from the physical standpoint.