  We study some two-dimensional dilaton gravity models using the formal theory
of partial differential equations. This allows us to prove that the reduced
phase space is two-dimensional without an explicit construction. By using a
convenient (static) gauge we reduce the theory to coupled \ode s and we are
able to derive for some potentials of interest closed-form solutions. We use an
effective (particle) Lagrangian for the reduced field equations in order to
quantize the system in a finite-dimensional setting leading to an exact partial
differential Wheeler-DeWitt equation instead of a functional one. A WKB
approximation for some quantum states is computed and compared with the
classical Hamilton-Jacobi theory. The effect of minimally coupled matter is
examined.
