Considering the lack of consensus for defining extreme events, we propose to revisit their definition. Our definition is based on the topological characterization of nominal dynamics, that is, hereafter, distinguished from extreme events with the help of two thresholds, one for the oscillation amplitude and one for the return time between two successive intersections with a surface of section. Two examples are investigated: one 3D jerk system and a 9D model for Rayleigh–Bénard convection. In the first example, the tipping point between small- and large-amplitude oscillations is topologically identified. The possibility of predicting the latter is investigated.