Climate extremes threaten human health, economic stability, and the safety of both natural and built environments. Compound extreme events are combinations of multiple climate drivers and/or hazards that contribute to societal or environmental risks, and their impacts on human society and natural ecosystems are often more serious and destructive than those of a single extreme event. Understanding the changes in compound extreme events is important for adaptation, mitigation strategies, and disaster risk management. Here, the definitions and connotations of compound extreme events are briefly discussed, including preconditioned, multivariate, temporal, and spatial compounding events. Subsequently, the progress in compound extreme event research is discussed in terms of temporal and spatial evolution characteristics, influencing factors, and future scenario projections. Given the problems in current research, we suggest that future studies should focus on studying compound extreme events regarding variable/index selection and threshold determination, dependence and interaction analysis among drivers and/or hazards, simulation performance evaluation and future projections, and their dynamic processes and disaster-causing mechanisms. Compound extreme events are expected to increase in frequency and intensity in a warming world, and many regions are projected to experience an increase in the probability of compound events with greater global warming. Therefore, we must improve our understanding of the causes and drivers of compound and cascade events.