Until recently, little was known about the Weddell Sea sector of the West Antarctic Ice Sheet. In the last 10 years, a variety of expeditions and numerical modelling experiments have improved knowledge of its glaciology, glacial geology and tectonic setting. Two of the sector's largest ice streams rest on a steep reverse-sloping bed yet, despite being vulnerable to change, satellite observations show contemporary stability. There is clear evidence for major ice sheet reconfiguration in the last few thousand years, however. Knowing precisely how and when the ice sheet has changed in the past would allow us to better understand whether it is now at risk. Two competing hypotheses have been established for this glacial history. In one, the ice sheet retreated and thinned progressively from its Last Glacial Maximum position. Retreat stopped at, or very near, the present position in the Late Holocene. Alternatively, in the Late Holocene, the ice sheet retreated significantly upstream of its present grounding line and then advanced to today's location due to glacial isostatic adjustment and ice shelf and ice rise buttressing. Both hypotheses point to data and theory in their support yet neither has been unequivocally tested or falsified. Here we review geophysical evidence to determine how each hypothesis has been formed, where there are inconsistencies in the respective glacial histories, how they may be tested or reconciled, and what new evidence is required to reach a common model for the Late Holocene ice sheet history of the Weddell Sea sector of West Antarctica.