  We derive and analyze high order discontinuous Galerkin methods for
second-order elliptic problems on implicitely defined surfaces in
$\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous
Galerkin framework of Arnold et al. [2002] on a triangulated surface
approximating the smooth surface. We prove optimal error estimates in both a
(mesh dependent) energy norm and the $L^2$ norm.
