  A degree-regular triangulation is one in which each vertex has identical
degree. Our main result is that any such triangulation of a (possibly
non-compact) surface $S$ is geometric, that is, it is combinatorially
equivalent to a geodesic triangulation with respect to a constant curvature
metric on $S$, and we list the possibilities. A key ingredient of the proof is
to show that any two $d$-regular triangulations of the plane for $d> 6 $ are
combinatorially equivalent. The proof of this uniqueness result, which is of
independent interest, is based on an inductive argument involving some
combinatorial topology.
