This paper investigates a class of systems with discontinuous right-hand side, which are widely used in applications. Discontinuous systems are closely related to the concept of differential inclusion, which was first introduced by A. Marchaud and S.K. Zaremba. Three different approaches to the definition of differential inclusions are presented: the Filippov, the Aizerman–Pyatnitskiy, and the Gelig–Leonov–Yakubovich definitions. For the class of systems considered, it is shown when these definitions coincide and when they are different.