  In this paper we provide key estimates used in the stability and error
analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains
with curved boundaries. In particular, we review trace estimates, inverse
estimates, discrete Poincar\'e--Friedrichs' inequalities, and optimal
interpolation estimates in noninteger Hilbert-Sobolev norms, that are well
known in the case of polytopal domains. We also prove curvature bounds for
curved simplices, which does not seem to be present in the existing literature,
even in the polytopal setting, since polytopal domains have piecewise zero
curvature. We demonstrate the value of these estimates, by analysing the IPDG
method for the Poisson problem, introduced by Douglas and Dupont
[\emph{Computing Methods in Applied Sciences, Lecture Notes in Physics, vol 58.
Springer, Berlin, Heidelberg}, pages 207--216. Springer, 1976], and by
analysing a variant of the $hp$-DGFEM for the biharmonic problem introduced by
Mozolevski and S\"{u}li [\emph{Computer Methods in Applied Mechanics and
Engineering}, 196(13-16):1851--1863, 2007]. In both cases we prove stability
estimates and optimal a priori error estimates. Numerical results are provided,
validating the proven error estimates.
