  We study the properties of CAT($\kappa$) surfaces: length metric spaces
homeomorphic to a surface having curvature bounded above in the sense of
satisfying the CAT($\kappa$) condition locally. The main facts about
CAT($\kappa$) surfaces seem to be largely a part of mathematical folklore, and
this paper is intended to rectify the situation. We provide three distinct
proofs of the fact that CAT($\kappa$}) surfaces have bounded integral
curvature. This fact allows us to use the established theory of surfaces of
bounded curvature to derive further properties of CAT($\kappa$) surfaces. Among
other results, we show that such surfaces can be approximated by smooth
Riemannian surfaces of Gaussian curvature at most $\kappa$. We do this by
giving explicit formulas for smoothing the vertices of model polyhedral
surfaces.
