Classical and quantum gravity in 1+1 dimensions .3. Solutions of arbitrary topology

All global solutions of arbitrary topology of the most general (1 + 1)-dimensional dilaton gravity models are obtained. We show that for a generic model there are globally smooth solutions on any non-compact 2-surface. The solution space is parametrized explicitly and the geometrical significance of continuous and discrete labels is elucidated. As a corollary we gain insight into the (in general non-trivial) topology of the reduced phase space. The classification covers basically all two-dimensional metrics of Lorentzian signature with a (local) Killing symmetry.