Classical and quantum gravity in dimensions: III. 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 (reasonable) non-compact topology, e.g. on any punctured two-surface. For higher genus they are necessarily connected with gravitational kinks, i.e. with a twisted lightcone-structure. The solution space is parametrized explicitly and the geometrical significance of continuous and discrete labels is elucidated. As a corollary we determine the (in general non-trivial) topology of the reduced phase space. The classification covers basically all 2D metrics of Lorentzian signature with a (local) Killing symmetry.