Vinberg representations are representations of algebraic groups that arise from a cyclic grading of a semisimple Lie algebra. In the literature, they are mainly known as theta-groups or Vinberg pairs. A distinguishing feature of these representations is that it is possible to classify the orbits of the algebraic group. We sketch how this can be done when the base field is the complex numbers. This mainly uses results of Vinberg of the 1970s. Then we describe techniques for classifying the orbits when the base field is the real numbers. This talk is based on joint work with Mikhail Borovoi, Hong Van Le, Heiko Dietrich, Marcos Origlia, and Alessio Marrani (part of the activity of the MIUR Excellence Department Project Mat-Mod@TOV (CUP E83C23000330006)).