Environmental extreme events can have devastating impacts on society when they interact with vulnerable human and natural systems. Such events can result from natural causes, like phenomena related to the El Ni~no-Southern Oscillation or decadal/multi-decadal climate variations. These causes can follow an increase in human activity, e.g., through land-use changes or anthropogenic climate change, that can influence the frequency, intensity, spatial extent and timing of these events, and spur unprecedented extremes. To accurately understand and quantify the risks associated with these events, it is important to identify trends related to these causes, which may be measured or unmeasured.

Fitting models for rare events is inherently difficult because of the paucity of data available. The most destructive extreme events are rarely isolated in space and time, so one must account for their spatial and temporal dependencies. This thesis deals with the parametric modelling of severe thunderstorms and wildfires using models motivated from limiting probabilistic results.

The first part of this thesis explores influences on the magnitude and spatial extent of extremes of environments related to severe US thunderstorms. Our results show that the risk from severe thunderstorms in April and May is increasing in parts of the US where it was already high, and that the risk from storms in February increases during La Ni~na years. We also show that these extremes are more localized during spring/summer seasons than in the winter, and find that some of these seasonal differences are more pronounced during El Ni~no years.

The second part of the thesis deals with predicting and explaining the spatial extent, frequency, intensity and timing of wildfires using meteorological and land-use covariates. Our first approach uses ideas from extreme-value theory in a machine learning context to give good prediction of the distributional tails of our data. The second approach uses a novel Bayesian hierarchical model designed specifically for extreme wildfires. We show that wildfire risk on the French Mediterranean basin is affected by significant random effects related to land-use and policy changes, and a seasonally-varying fire-weather index.