We study the properties of CAT(κ) surfaces: length metric spaces
homeomorphic to a surface having curvature bounded above in the sense of
satisfying the CAT(κ) condition locally. The main facts about
CAT(κ) 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(κ}) 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(κ) surfaces. Among
other results, we show that such surfaces can be approximated by smooth
Riemannian surfaces of Gaussian curvature at most κ. We do this by
giving explicit formulas for smoothing the vertices of model polyhedral
surfaces.Comment: 23 pages, 3 figure