Statistics of extremes deals with the estimation of events that have low probabilities but potentially large consequences, such as stock market gyrations, flooding and heatwaves. Very often the events of interest have never yet been observed, and hence their probabilities must be estimated by extrapolation well outside any existing data, using specially adapted for the purpose. This area exhibits a beautiful interplay between probability and statistics, has a long history and a rich tradition of applications, originally in insurance and engineering, but increasingly in the environmental sciences and in finance. Statistical methods for modelling scalar and multivariate extremes based on sample maxima and threshold exceedances are well-established, but recently the focus has turned to more complex settings, including spatial and space-time modelling of extremal phenomena. In this lecture I shall survey recent work on the topic and then show how detailed modelling for high-dimensional problems can be undertaken using Pareto processes, generalised versions of threshold exceedances and gradient scoring rules. The work is joint with colleagues too numerous to list here.