Extreme events, such as rogue waves, earthquakes, and stock market crashes, occur spontaneously in many dynamical systems. Because of their usually adverse consequences, quantification, prediction, and mitigation of extreme events are highly desirable. Here, we review several aspects of extreme events in phenomena described by high-dimensional, chaotic dynamical systems. We especially focus on two pressing aspects of the problem: (i) mechanisms underlying the formation of extreme events and (ii) real-time prediction of extreme events. For each aspect, we explore methods relying on models, data, or both. We discuss the strengths and limitations of each approach as well as possible future research directions.