Sanders and Gyakum's (1980) 24-hour criterion (a central pressure fall that averages at least 1 mb h−1 for 24 h) of a rapidly deepening extratropical surface cyclone is adopted to investigate explosive surface cyclogenesis associated with cyclones in polar air streams and frontal waves induced by cyclones in polar air streams. A case is described in detail in which both the cold air vortex and frontal wave meet the criterion. Other examples of rapid cyclogenesis in which only the cold air cyclone or frontal wave satisfy the criterion are given. These examples confirm that cyclones in polar streams and frontal waves induced by cyclones in polar air streams can exhibit rapid enough deepening rates to fulfill Sanders and Gyakum's criterion. These events are located downstream of a mobile upper-tropospheric trough, within or poleward of the jet stream axis, and within or ahead of a planetary-wave trough. A quasi-geostrophic diagnosis of the case described in detail reveals computed pressure falls much less than those observed. It appears likely that quasi-geostrophic deepening rates for the other examples would also be far short of observed rates. These findings are in accordance with Sanders and Gyakum's (1980) observations.