Abstract. We introduce Priority Channel Systems, a new natural class of channel systems where messages carry a numeric priority and where higher-priority messages can supersede lower-priority messages preceding them in the fifo communication buffers. The decidability of safety and inevitability properties is shown via the introduction of a priority embedding, a well-quasi-ordering that has not previously been used in well-structured systems. We then show how Priority Channel Systems can compute fast-growing functions and prove that the aforementioned verification problems are Fε 0 -complete.