High latency and loss rates in the Internet make it difficult to stream audio and video without introducing a large playback delay. To address these problems, we propose a prefix caching technique whereby a proxy stores the initial frames of popular clips. Upon receiving a request for the stream, the proxy initiates transmission to the client and simultaneously requests the remaining frames from the server. In addition to hiding the delay, throughput, and loss effects of a weaker service model between the server and the proxy, this novel yet simple prefix caching technique aids the proxy in performing workahead smoothing into the client playback buffer. By transmitting large frames in advance of each burst, workahead smoothing substantially reduces the peak and variability of the network resource requirements along the path from the proxy to the client. We describe how to construct a smooth transmission schedule, based on the size of the prefix, smoothing, and playback buffers, without increasing client playback delay. Experiments with MPEG traces show how a few megabytes of buffer space at the proxy can substantially reduce the bandwidth requirements of variable-bit-rate video. Drawing on these results, we present guidelines for allocating buffer space for each stream, and how to effectively share buffer and bandwidth resources among multiple clients and streams.