Summary: It has long been recognised that standard process algebra has difficulty dealing with actions of different priority, such as for instance an interrupt action of high priority. Various solutions have been proposed. We introduce a new approach, involving the addition of “priority guards” to Milner’s process calculus CCS. In our approach, priority is unstratified, meaning that actions are not assigned fixed levels, so that the same action can have different priority depending where it appears in a program. Unlike in other unstratified accounts of priority in CCS (such as that of Camilleri and Winskel), we treat inputs and outputs symmetrically. We introduce the new calculus, give examples, develop its theory (including bisimulation and equational laws), and compare it with existing approaches. We use leader election problems to show that priority adds expressiveness to both CCS and the \(\pi \)-calculus.