This paper presents an algorithm for performing on-the-fly race detection for parallel message-passing programs. The algorithm reads a trace of the communication events in a message-passing parallel program and either finds a specific race condition or reports that the traced program is race-free. It supports a rich message-passing model, including blocking and non-blocking sends and receives, synchronous and asynchronous sends, receive selectivity by source and/or tag value, and arbitrary amounts of system buffering of messages. It runs in polynomial time and is very efficient for most types of executions. A key feature of the race detection algorithm is its use of several new types of logical clocks for determining ordering relations. It is likely that these logical clocks will also be useful in other settings.