  In any quantum theory of gravity, it is of the utmost importance to recover
the limit of quantum theory in an external spacetime. In quantum
geometrodynamics (quantization of general relativity in the Schr\"odinger
picture), this leads in particular to the recovery of a semiclassical (WKB)
time which governs the dynamics of non-gravitational fields in spacetime. Here,
we first review this procedure with special emphasis on conceptual issues. We
then turn to an alternative theory - Weyl (conformal) gravity, which is defined
by a Lagrangian that is proportional to the square of the Weyl tensor. We
present the canonical quantization of this theory and develop its semiclassical
approximation. We discuss in particular the extent to which a semiclassical
time can be recovered and contrast it with the situation in quantum
geometrodynamics.
