More than a decade ago, ice core records from Greenland revealed that the last glacial period was characterized by abrupt climate changes that recurred on millennial time scales. Since their discovery, there has been a large effort to determine whether these climate events were a global phenomenon or were just confined to the North Atlantic region and also to reveal the mechanisms that were responsible for them. In this paper, we review the available paleoclimate observations of abrupt change during the last glacial period in order to place constraints on possible mechanisms. Three different mechanisms are then reviewed: ocean thermohaline circulation, sea ice feedbacks, and tropical processes. Each mechanism is tested for its ability to explain the key features of the observations, particularly with regard to the abruptness, millennial recurrence, and geographical extent of the observed changes. It is found that each of these mechanisms has explanatory strengths and weaknesses, and key areas in which progress could be made in improving the understanding of their long-term behavior, both from observational and modeling approaches, are suggested. Finally, it is proposed that a complete understanding of the mechanisms of abrupt change requires inclusion of processes at both low and high latitudes, as well as the potential for feedbacks between them. Some suggestions for experimental approaches to test for such feedbacks with coupled climate models are given.