Linear systems on complex algebraic curves have been an interesting and rich area of study for a long time. It is known that a real structure on the initial curve induces a real structure on the varieties parameterizing these linear systems. In this talk, I will first present the classical Brill-Noether theorems for these varieties, and then address some fundamental topological questions in the real setting including the case of real trigonal curves. This talk is based on the work https://arxiv.org/abs/2412.04115 .