•A review of AMOC stability in ocean alone and coupled climate models.•Comments on the very frequently used AMOC stability criterion.•Comments on the use of flux adjusted climate models.•Comments on AMOC stability in eddy-resolving versus non-eddy-resolving ocean components. The stability of the Atlantic meridional overturning circulation (AMOC) in ocean models depends quite strongly on the model formulation, especially the vertical mixing, and whether it is coupled to an atmosphere model. A hysteresis loop in AMOC strength with respect to freshwater forcing has been found in several intermediate complexity climate models and in one fully coupled climate model that has very coarse resolution. Over 40% of modern climate models are in a bistable AMOC state according to the very frequently used simple stability criterion which is based solely on the sign of the AMOC freshwater transport across 33° S. In a recent freshwater hosing experiment in a climate model with an eddy-permitting ocean component, the change in the gyre freshwater transport across 33° S is larger than the AMOC freshwater transport change. This casts very strong doubt on the usefulness of this simple AMOC stability criterion. If a climate model uses large surface flux adjustments, then these adjustments can interfere with the atmosphere–ocean feedbacks, and strongly change the AMOC stability properties. AMOC can be shut off for many hundreds of years in modern fully coupled climate models if the hosing or carbon dioxide forcing is strong enough. However, in one climate model the AMOC recovers after between 1000 and 1400 years. Recent 1% increasing carbon dioxide runs and RCP8.5 future scenario runs have shown that the AMOC reduction is smaller using an eddy-resolving ocean component than in the comparable standard 1° ocean climate models.