A challenging topic in statistics concerns inferring about extreme events; not necessarily the upper quartile event, but those way out in the tails of the distribution. The problem is essentially one of having to extrapolate far beyond the range of the data, and while such a procedure is generally not recommended, the importance of the problem demands it be done anyway. Extreme-value theory (EVT) provides justification for utilizing certain distributions in this context, namely, in the univariate context: the generalized extreme value (GEV) for maxima taken over very long stretches of time (e.g. annual) and the generalized Pareto (GP) for excesses over a very high threshold. The latter approximates the tail of the former, and both encompass three types of distributions according to the value of their shape parameters. A third, point process, characterization ties the two approaches together. Particularly for environmental topics, incorporating spatial information into any such analysis is important, but although multivariate methods exist in EVT, analyzing spatial data for extremes is a tricky, and active, area of research. This talk will review these methods.