AbstractPeaks-over-threshold analysis using the generalised Pareto distribution is widely applied in modelling tails of univariate random variables, but much information may be lost when complex extreme events are studied using univariate results. In this paper, we extend peaks-over-threshold analysis to extremes of functional data. Threshold exceedances defined using a functional r are modelled by the generalised r-Pareto process, a functional generalisation of the generalised Pareto distribution that covers the three classical regimes for the decay of tail probabilities, and that is the only possible continuous limit for r-exceedances of a properly rescaled process. We give construction rules, simulation algorithms and inference procedures for generalised r-Pareto processes, discuss model validation and apply the new methodology to extreme European windstorms and heavy spatial rainfall.