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Extreme space weather events, while rare, can have a substantial impact on our technologically-dependent society. And, although such events have only occasionally been observed, through careful analysis of a wealth of space-based and ground-based observations, historical records, and extrapolations from more moderate events, we have developed a basic picture of the components required to produce them. Several key issues, however, remain unresolved. For example, what limits are imposed on the maximum size of such events? What are the likely societal consequences of a so-called "100-year" solar storm? In this review, we summarize our current scientific understanding about extreme space weather events as we follow several examples from the Sun, through the solar corona and inner heliosphere, across the magnetospheric boundary, into the ionosphere and atmosphere, into the Earth's lithosphere, and, finally, its impact on man-made structures and activities, such as spacecraft, GPS signals, radio communication, and the electric power grid. We describe preliminary attempts to provide probabilistic forecasts of extreme space weather phenomena, and we conclude by identifying several key areas that must be addressed if we are better able to understand, and, ultimately, predict extreme space weather events. [ABSTRACT FROM AUTHOR]