A natural question in the study of semigroups of operators is to characterize their generators. Recently, generators of semigroups of composition operators were characterized in the context of a large variety of Banach spaces of analytic functions in the unit disc by Arendt and Chalendar and, independently, in a joint work with D. Yakubovich. In this talk, we will discuss these results along with the recent ones regarding $C_0$-semigroups of weighted composition operators obtained in collaboration with A. Siskakis and D. Yakubovich.