  The paper develop the alternative formulation of quantum mechanics known as
the phase space quantum mechanics or deformation quantization. It is shown that
the quantization naturally arises as an appropriate deformation of the
classical Hamiltonian mechanics. More precisely, the deformation of the
point-wise product of observables to an appropriate noncommutative
$\star$-product and the deformation of the Poisson bracket to an appropriate
Lie bracket is the key element in introducing the quantization of classical
Hamiltonian systems.
  The considered class of deformations and the corresponding $\star$-products
contains as a special cases all deformations which can be found in the
literature devoted to the subject of the phase space quantum mechanics.
  Fundamental properties of $\star$-products of observables, associated with
the considered deformations are presented as well. Moreover, a space of states
containing all admissible states is introduced, where the admissible states are
appropriate pseudo-probability distributions defined on the phase space. It is
proved that the space of states is endowed with a structure of a Hilbert
algebra with respect to the $\star$-multiplication.
  The most important result of the paper shows that developed formalism is more
fundamental then the axiomatic ordinary quantum mechanics which appears in the
presented approach as the intrinsic element of the general formalism. The
equivalence of two formulations of quantum mechanics is proved by observing
that the Wigner-Moyal transform has all properties of the tensor product. This
observation allows writing many previous results found in the literature in a
transparent way, from which the equivalence of the two formulations of quantum
mechanics follows naturally.
