Abstract : It is over 40 years since the first seminal work on priority assignment for real-time systems using fixed priority scheduling. Since then, huge progress has been made in the field of real-time scheduling with more complex models and schedulability analysis techniques developed to better represent and analyse real systems. This tutorial style review provides an in-depth assessment of priority assignment techniques for hard real-time systems scheduled using fixed priorities. It examines the role and importance of priority in fixed priority scheduling in all of its guises, including: pre-emptive and non-pre-emptive scheduling; covering single- and multi-processor systems, and networks. A categorisation of optimal priority assignment techniques is given, along with the conditions on their applicability. We examine the extension of these techniques via sensitivity analysis to form robust priority assignment policies that can be used even when there is only partial information available about the system. The review covers priority assignment in a wide variety of settings including: mixed-criticality systems, systems with deferred pre-emption, and probabilistic real-time systems with worst-case execution times described by random variables. It concludes with a discussion of open problems in the area of priority assignment.