<p>The present-day Atlantic Ocean circulation is susceptible to large-scale instabilities that, if they develop, would have worldwide climate impacts. Transitions between circulation patterns owing to such instabilities have been found across a hierarchy of ocean-climate models, but it remains difficult to determine whether they will occur under future climate change. We discuss how a dynamical systems approach can be used to identify basic destabilization mechanisms, determine the essential transition behaviour and thereby unify much of the model behaviour that has been found. This approach helps to interpret the complex behaviour seen in climate records and existing model simulations, to design new climate model experiments and, eventually, to quantitatively assess the stability of the Atlantic Ocean circulation under present-day and future climate forcing.</p>