  A classification of the maximally extended solutions for 1+1 gravity models
(comprising e.g. generalized dilaton gravity as well as models with non-trivial
torsion) is presented. No restrictions are placed on the topology of the
arising solutions, and indeed it is found that for generic models solutions on
non-compact surfaces of arbitrary genus with an arbitrary non-zero number of
holes can be obtained. The moduli space of classical solutions (solutions of
the field equations with fixed topology modulo gauge transformations) is
parametrized explicitly.
