Abstract: I will discuss recent results on the mathematical study of active Brownian particle systems, a popular paradigmatic system for self-propelled particles. I will present four microscopic models with different types of repulsive interactions between particles and their associated macroscopic equations, which are formally obtained using different coarse-graining methods. The macroscopic limits are integro-differential equations for the density in phase space (positions and orientations) of the particles and may include nonlinearities in both the diffusive and advective components. In contrast to passive particles, systems of active particles can undergo phase separation without any attractive interactions, a mechanism known as motility-induced phase separation (MIPS). This aspect will be discussed for each model in the parameter space of occupied volume fraction and Péclet number via a linear stability analysis and numerical simulations. In the second part of the seminar I will focus on the well-posedness for one of the model previously mentioned. More precisely, I will present our strategy to provide existence, uniqueness, and continuous regularity, which might be applied to a larger class of space-periodic parabolic equations presenting nonlocal mobility. The seminar is based on works in collaboration with M. Bruna (Cambridge), M. Burger (Erlangen), and S. M. Schulz (Wisconsin-Madison).