  This is a chapter of a forthcoming Lecture Notes in Mathematics "Modern
Approaches to Discrete Curvature" edited by L. Najman and P. Romon. It provides
a survey on geometric and spectral consequences of curvature bounds. The
geometric setting are tessellations of surfaces with finite and vanishing
genus. We consider a curvature arising as an angular defect. Several of the
results presented here have analogues in Riemannian geometry. In some cases one
can go even beyond the Riemannian results and there also striking differences
which shall be highlighted.
