  The existing approaches to quantization of gravity aim at giving quantum
description of 3-geometry following to the ideas of the Wheeler -- DeWitt
geometrodynamics. In this description the role of gauge gravitational degrees
of freedom is missed. A probable alternative is to consider gravitational
dynamics in extended phase space, taking into account the distinctions between
General Relativity and other field theories. The formulation in extended phase
space leads to some consequences at classical and quantum levels. At the
classical level, it ensures that Hamiltonian dynamics is fully equivalent to
Lagrangian dynamics, and the algebra of Poisson brackets is invariant under
reparametrizations in a wide enough class including reparametrizations of gauge
variables, meantime in the canonical Dirac approach the constraints' algebra is
not invariant that creates problems with quantization. At the quantum level,
the approach come to the description in which the observer can see various but
complementary quantum gravitational phenomena in different reference frames
that answers the spirit of General Relativity and Quantum Theory. Though until
now the approach was applied to General Relativity in its original
formulations, its implementation in different trends, including Quantum Loop
Gravity or some other representations of gravitational variables, would also be
of interest.
