  The review of star-product formalism providing the possibility to describe
quantum states and quantum observables by means of the functions called symbols
of operators which are obtained by means of bijective maps of the operators
acting in Hilbert space onto these functions is presented. Examples of the
Wigner-Weyl symbols (like Wigner quasi-distributions) and tomographic
probability distributions (symplectic, optical and photon-number tomograms)
identified with the states of the quantum systems are discussed. Properties of
quantizer-dequantizer operators which are needed to construct the bijective
maps of two operators (quantum observables) onto the symbols of the operators
are studied. The relation of the structure constants of the associative
star-product of the operator symbols to the quantizer-dequantizer operators is
reviewed.
