The ''problem of time'' in canonical quantum gravity refers to the dif
ficulties involved in defining a Hilbert space structure on states, an
d local observables on this Hilbert space, for a theory in which the s
pacetime metric is treated as a quantum field, so no classical metrica
l or causal structure is present on spacetime. We describe an approach
, much in the spirit of ideas proposed by Misner, Kuchar, and others,
to defining states and local observables in quantum gravity which expl
oits the analogy between the Hamiltonian formulation of general relati
vity and that of a relativistic particle. In the case of minisuperspac
e models, a concrete theory is obtained which appears to be mathematic
ally and physically viable, although it contains some radical features
with regard to the presence of an ''arrow of time.'' The viability of
this approach in the case of infinitely many degrees of freedom rests
on a number of fairly well-defined issues, which, however, remain unr
esolved. As a by-product of our analysis, the theory of a relativistic
particle in curved spacetime is developed.