The paper is the first of two parts of a work reviewing some approaches to
the problem of time in quantum cosmology, which were put forward last decade,
and which demonstrated their relation to the problems of reparametrization and
gauge invariance of quantum gravity. In the present part we remind basic
features of quantum geometrodynamics and minisuperspace cosmological models,
and discuss fundamental problems of the Wheeler - DeWitt theory. Various
attempts to find a solution to the problem of time are considered in the
framework of the canonical approach. Possible solutions to the problem are
investigated making use of minisuperspace models, that is, systems with a
finite number of degrees of freedom. At the same time, in the last section of
the paper we expand our consideration beyond the minisuperspace approximation
and briefly review promising ideas by Brown and Kuchar, who propose that dust
interacting only gravitationally can be used for time measuring, and the
unitary approach by Barvinsky and collaborators. The latter approach admits
both the canonical and path integral formulations and anticipates the
consideration of recent developments in the path integral approach in the
second part of our work.Comment: 16 pages, to be published in Grav. Cosmo
