  We extend the applicability of the popular interior-penalty discontinuous
Galerkin (dG) method discretizing advection-diffusion-reaction problems to
meshes comprising extremely general, essentially arbitrarily-shaped element
shapes. In particular, our analysis allows for \emph{curved} element shapes,
without the use of non-linear elemental maps. The feasibility of the method
relies on the definition of a suitable choice of the discontinuity
penalization, which turns out to be explicitly dependent on the particular
element shape, but essentially independent on small shape variations. This is
achieved upon proving extensions of classical trace and Markov-type inverse
estimates to arbitrary element shapes. A further new $H^1-L_2$-type inverse
estimate on essentially arbitrary element shapes enables the proof of inf-sup
stability of the method in a streamline-diffusion-like norm. These inverse
estimates may be of independent interest. A priori error bounds for the
resulting method are given under very mild structural assumptions restricting
the magnitude of the local curvature of element boundaries. Numerical
experiments are also presented, indicating the practicality of the proposed
approach.
