  We develop novel stabilized cut discontinuous Galerkin (CutDG) methods for
advection-reaction problems. The domain of interest is embedded into a
structured, unfitted background mesh in $\mathbb{R}^d$ where the domain
boundary can cut through the mesh in an arbitrary fashion. To cope with
robustness problems caused by small cut elements, we introduce ghost penalties
in the vicinity of the embedded boundary to stabilize certain (semi)-norms
associated with the advection and reaction operator. A few abstract assumptions
on the ghost penalties are identified enabling us to derive geometrically
robust and optimal a priori error and condition number estimates for the
stationary advection-reaction problem which hold irrespective of the particular
cut configuration. Possible realizations of suitable ghost penalties are
discussed. The theoretical results are corroborated by a number of
computational studies for various approximation orders and for two and
three-dimensional test problems.
