  We summarize the main results of our investigation of B-type topological
Landau-Ginzburg models whose target is an arbitrary open Riemann surface. Such
a Riemann surface need not be affine algebraic and in particular it may have
infinite genus or an infinite number of Freudenthal ends. Under mild conditions
on the Landau-Ginzburg superpotential, we give a complete description of the
triangulated structure of the category of topological D-branes in such models
as well as counting formulas for the number of topological D-branes considered
up to relevant equivalence relations.
