In this paper we offer alternative axioms of quantum mechanics which we consider to have a better physical basis. To each quantum system a separable unital C*-algebra is assigned, the observables are represented by hermitian elements of that algebra, while the states are represented as positive linear functionals of norm one. First we analyzed necessary conceptual changes in the development of quantum mechanics that lie in the roots of the formalism. By defining observables and states operationally, we build a formalism suitable for a description of a general physical system. We present the basic concepts and results from the theory of C*-algebras. We show that by inserting additional assumptions - the possibility of a simultaneous and precise determination of all observables - leads to the formalism of classical mechanics. We show that classical systems are represented by commutative C*-algebras. We then present the main results from the theory of representations of C*-algebras: GNS construction, Gelfand-Naimark’s theorem, and the connection between irreducible representations and pure states. From this we deduce the basic Dirac-von Neumann axioms. Finally, we studied the system’s dynamics. It is shown that it can be described by one-parameter groups of *-automorphisms and with the aid of the Stone’s theorem we derive the Schrödinger’s equation.