Many natural, technological, or social systems are capable of recurrently generating large impact events. Well known are earthquakes, tsunamis, or extreme weather events — such as heat waves, droughts, floods, heavy precipitation, or tornadoes — that can lead to disasters when interacting with exposed or vulnerable human or natural systems. We define an extreme event as a rare and recurrent event, during which an appropriate variable exhibits an unusual behavior, e.g., possesses an extremely large or small value. Simple examples of appropriate observables in physical systems would be the wave height for oceanic waves or the intensity of the optical output for optical rogue waves. Less simple observables, would be the abundance of a toxic algal species in a harmful algal bloom, since it may not be the most abundant species in the plankton community, but have the largest impact on the ecosystem due to their toxin production. Another example would be the level of synchrony of populations of neurons in the brain, which could be considered as an appropriate indication for an epileptic seizure, which is an extreme event for the affected individual. In this project, we investigate the mechanisms which lead to the generation and termination of such extreme events in spatially extended excitable systems. The focal areas of this research are the harmful algal blooms and epileptic seizures.