A long-tailed or heavy-tailed probability distribution is one that assigns relatively high probabilities to regions far from the mean or median. A more formal mathematical definition is given below. In the context of teletraffic engineering a number of quantities of interest have been shown to have a long-tailed distribution. For example, if we consider the sizes of files transferred from a web-server, then, to a good degree of accuracy, the distribution is heavy-tailed, that is, there are a large number of small files transferred but, crucially, the number of very large files transferred remains a major component of the volume downloaded. Many processes are technically long-range dependent but not self-similar. The differences between these two phenomena are subtle. Heavy-tailed refers to a probability distribution, and long-range dependent refers to a property of a time series and so these should be used with care and a distinction should be made. The terms are distinct although superpositions of samples from heavy-tailed distributions aggregate to form long-range dependent time series. Additionally there is Brownian motion which is self-similar but not long-range dependent. Overview The design of robust and reliable networks and network services has become an increasingly challenging task in today's Internet world. To achieve this goal, understanding the characteristics of Internet traffic plays a more and more critical role. Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic. Self-similar Ethernet traffic exhibits dependencies over a long range of time scales. This is to be contrasted with telephone traffic which is Poisson in its arrival and departure process. With many time-series if the series is averaged then the data begins to look smoother. However, with self-similar data, one is confronted with traces which are spiky and bursty, even at large scales. Such behaviour is caused by strong dependence in the data: large values tend to come in clusters, and clusters of clusters, etc. This can have far-reaching consequences for network performance. Heavy-tail distributions have been observed in many natural phenomena including both physical and sociological phenomena. Mandelbrot established the use of heavy-tail distributions to model real-world fractal phenomena, e.g. Stock markets, earthquakes, and the weather. Ethernet, WWW, SS7, TCP, FTP, TELNET and VBR video (digitised video of the type that is transmitted over ATM networks) traffic is self-similar. Self-similarity in packetised data networks can be caused by the distribution of file sizes, human interactions and/or