General relativity leads to dynamical systems which have a Hamiltonian flow in time together with a symmetry (time reparameterization invariance) that allows one to transform the time parameter by an arbitrary monotonic function. In the usual treatment of classical mechanics, one would apply symplectic reduction to the symmetry, but the Hamiltonian flow on the resulting quotient space is always trivial. This talk introduces "gauge fixing" as a method to define a non-trivial dynamical flow on a reparameterization invariant system. An algebraic approach to quantization then implies corresponding conditions on the algebra of observables and its states.