Compound weather and climate events describe combinations of multiple climate drivers and/or hazards that contribute to societal or environmental risk. Although many climate-related disasters are caused by compound events, the understanding, analysis, quantification and prediction of such events is still in its infancy. In this Review, we propose a typology of compound events and suggest analytical and modelling approaches to aid in their investigation. We organize the highly diverse compound event types according to four themes: preconditioned, where a weather-driven or climate-driven precondition aggravates the impacts of a hazard; multivariate, where multiple drivers and/or hazards lead to an impact; temporally compounding, where a succession of hazards leads to an impact; and spatially compounding, where hazards in multiple connected locations cause an aggregated impact. Through structuring compound events and their respective analysis tools, the typology offers an opportunity for deeper insight into their mechanisms and impacts, benefiting the development of effective adaptation strategies. However, the complex nature of compound events results in some cases inevitably fitting into more than one class, necessitating soft boundaries within the typology. Future work must homogenize the available analytical approaches into a robust toolset for compound-event analysis under present and future climate conditions.