In this paper we study hard real-time systems: systems where strict time deadlines have to be met. We analyze a special case as well as a general model for hard real-time systems and study pre-emptive, static, scheduling policies for a single processor. The analysis is exact and can handle any arbitrary choice of strict deadlines. For any specification of a hard real-time system, a feasible priority assignment is one where all deadlines are met. An optimal scheduling algorithm is an algorithm that always produces a feasibile priority assignment if one exists. For both the special and general model we present an optimal scheduling algorithm.