It has long been recognised that ordinary 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 the summation operator of 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. An important feature is that, unlike in other unstratified accounts of priority in CCS (such as that of Camilleri and Winskel), we can treat inputs and outputs symmetrically. We introduce the new calculus, give examples, develop its theory (including bisimulation, equational laws and logics), and compare it with existing approaches.