Over 40 years ago, Raup and Sepkoski identified five episodes of elevated extinction in the marine fossil record that were thought to be statistically distinct, thus warranting the term the “Big Five” mass extinctions. Since then, the term has become part of standard vocabulary, especially with the naming of the current biodiversity crisis as the “sixth mass extinction.” However, there is no general agreement on which time intervals should be viewed as mass extinctions, in part because the Big Five turn out not to be statistically distinct from background rates of extinction, and in part, because other intervals of time have even higher extinction rates, in the Cambrian and early Ordovician. Nonetheless, the Big Five represent the five largest events since the early Ordovician, including in analyses that attempt to compensate for the incompleteness of the fossil and rock records. In the last 40 years, we have learned a great deal about the causes of many of the major and minor extinction events and are beginning to unravel the mechanisms that translated the initial environmental disturbances into extinction. However, for many of the events, further understanding will require going back to the outcrop, where the patchy distribution of environments and pervasive temporal gaps in the rock record challenge our ability to establish true extinction patterns. As for the current biodiversity crisis, there is no doubt that the rate of extinction is among the highest ever experienced by the biosphere, perhaps the second highest after the end-Cretaceous bolide impact. However (and fortunately), the absolute number of extinctions is still relatively small – there is still time to prevent this becoming a genuine mass extinction. Given the arbitrariness of calling out the Big Five, perhaps the current crisis should be called the “incipient Anthropocene mass extinction” rather than the “sixth mass extinction.”