The explosion of the World Wide Web as a medium for information dissemination has made it important to understand its characteristics, in particular the distribution of its file sizes. This paper presents evidence that a number of file size distributions in the Web exhibit heavy tails, including files requested by users, files transmitted through the network, transmission durations of files, and files stored on servers. In addition, we argue that because of the presence of caching in the Web, the size distribution of transmitted files is primarily determined by the distribution of files available on the Web, and is relatively insensitive to the distribution of files requested by users. Finally, we discuss some of the implications of heavy-tailed transmission durations and relate these results to self-similarity in network traffic.