Abstract Scheduling problems with processing set restrictions have been studied extensively by computer scientists and operations researchers under different names. These include “scheduling typed task systems,” “multi-purpose machine scheduling,” “scheduling with eligibility constraints,” “scheduling with processing set restrictions,” and “semi-matchings for bipartite graphs.” In this paper we survey the state of the art of these problems. Our survey covers offline and online problems, as well as nonpreemptive and preemptive scheduling environments. Our emphasis is on polynomial-time algorithms, complexity issues, and approximation schemes. While our main focus is on the makespan objective, other performance criteria are also discussed.