  This paper is an extended version of the talk given at 19th Texas Symposium
of Relativistic Astrophysics and Cosmology, Paris, 1998. It reviews of some
recent work; mathematical details are skipped. It is well-known that a choice
of gauge in generally covariant models has a twofold pupose: not only to render
the dynamics unique, but also to define the spacetime points. A geometric way
of choosing gauge that is not based on coordinate conditions---the so-called
covariant gauge fixing---is described. After a covariant gauge fixing, the
dynamics is unique and the background manifold points are well-defined, but the
description remains invariant with respect to all diffeomorphisms of the
background manifold. Transformations between different covariant gauge fixings
form the well-known Bergmann-Komar group. Each covariant gauge fixing
determines a so-called Kucha\v{r} decomposition. The construction of the
quantum theory is based on the Kucha\v{r} form of the action and the Dirac
method of operator constraints. It is demonstrated that the Bergmann-Komar
group is too large to be implementable by unitary maps in the quantum domain.
