The most general dilaton gravity theory in 2 spacetime dimensions is c
onsidered. A Hamiltonian analysis is performed and the reduced phase s
pace, which is two dimensional, is explicitly constructed in a suitabl
e parametrization of the fields. The theory is then quantized via the
Dirac method in a functional Schrodinger representation. The quantum c
onstraints are solved exactly to yield the (spatial) diffeomorphism in
variant physical wave functionals for all theories considered. These w
ave functionals depend explicitly on the single configuration space co
ordinate as well as on the imbedding of space into spacetime (i.e. on
the choice of time).