Impacts of climate events can be particularly large when multiple climate extremes coincide. If the occurrence of these extremes is not independent, risk assessments need to take dependencies into account to avoid systematic biases. Overall, many climate-related impacts are associated with a complex combination of climate drivers that interact at various spatial and temporal scales. Understanding such so-called compound climate events requires close collaboration between (climate) scientists, statisticians, and experts on climate impacts. In this chapter, we guide the reader through a number of concepts that together will provide a comprehensive overview of the field of compound event research, its relation to risk and climate change assessments, and their relevance for specific research problems.