The notion that the Atlantic Meridional Overturning Circulation (AMOC) can have more than one stable equilibrium emerged in the 1980s as a powerful hypothesis to explain rapid climate variability during the Pleistocene. Ever since, the idea that a temporary perturbation of the AMOC—or a permanent change in its forcing—could trigger an irreversible collapse has remained a reason for concern. Here we review literature on the equilibrium stability of the AMOC and present a synthesis that puts our understanding of past and future AMOC behavior in a unifying framework. This framework is based on concepts from Dynamical Systems Theory, which has proven to be an important tool in interpreting a wide range of model behavior. We conclude that it cannot be ruled out that the AMOC in our current climate is in, or close to, a regime of multiple equilibria. But there is considerable uncertainty in the location of stability thresholds with respect to our current climate state, so we have no credible indications of where our present-day AMOC is located with respect to thresholds. We conclude by identifying gaps in our knowledge and proposing possible ways forward to address these gaps.