  We develop a novel cut discontinuous Galerkin (CutDG) method for stationary
advection-reaction problems on surfaces embedded in $\mathbb{R}^d$. The CutDG
method is based on embedding the surface into a full-dimensional background
mesh and using the associated discontinuous piecewise polynomials of order $k$
as test and trial functions. As the surface can cut through the mesh in an
arbitrary fashion, we design a suitable stabilization that enables us to
establish inf-sup stability, a priori error estimates, and condition number
estimates using an augmented streamline-diffusion norm. The resulting CutDG
formulation is geometrically robust in the sense that all derived theoretical
results hold with constants independent of any particular cut configuration.
Numerical examples support our theoretical findings.
