We develop a novel cut discontinuous Galerkin (CutDG) method for stationary advection–reaction problems on surfaces embedded in Rd. 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.