  The paper is devoted to some of the difficulties which the Wheeler - DeWitt
quantum geometrodynamics encountered, in particular, a strong mathematical
proof that this theory is gauge-invariant, the definition of the wave function
of the Universe through a path integral and the illegality of asymptotic
boundary conditions in quantum gravity, the derivation of the Wheeler - DeWitt
equation from the path integral and the equivalence of the Dirac quantization
scheme with other approaches, the problem of definition of physical states in
quantum gravity, possible realizations of the Everett concept of "relative
states". The problems are rarely discussed in the literature. They are related
with the guiding idea that quantum theory of gravity must gauge invariant. It
will lead to the question if it is possible to achieve this goal in a
mathematically consistent way.
