In this paper, we investigate the non-preemptive scheduling problem as it arises with single processor systems. We extend some previously published results concerning preemptive and non-preemptive scheduling over a single processor. We examine non-idling and idling scheduling issues. The latter are of particular relevance in the case of non-preemption. We first embark on analyzing non-idling scheduling. The optimality of the non-idling non-preemptive Earliest Deadline First scheduling policy is revisited. Then, we provide feasibility conditions in the presence of aperiodic or periodic traffic. Second, we examine the concept of idling scheduling, whereby a processor can remain idle in the presence of pending tasks. The non-idling non-preemptiv- e Earliest Deadline First scheduling policy is not optimal since it is possible to find feasible task sets for which this policy fails to produce a valid schedule. An optimal algorithm to find a valid schedule (if any) is presented and its complexity analyzed. This paper shows that preemptive and non-preemptive scheduling are closely related. However, non-preemptive scheduling leads to more complex problems when combined with idling scheduling.