  We present a classification of all global solutions for generalized 2D
dilaton gravity models (with Lorentzian signature). While for some of the
popular choices of potential-like terms in the Lagrangian, describing, e.g.,
string inspired dilaton gravity or spherically reduced gravity, the possible
topologies of the resulting spacetimes are restricted severely, we find that
for generic choices of these `potentials' there exist maximally extended
solutions to the field equations on all non-compact two-surfaces.
