The Atlantic Meridional Overturning Circulation (AMOC) is a crucial element of the Earth climate. It is a complex circulation system difficult to monitor and to model. There is considerable debate regarding its evolution over the last century as well as large uncertainty about its fate at the end of this century. We depict here the progress since the IPCC SROCC report, offering an update of its chapter 6.7. We also show new results from a high-resolution ocean model and a CMIP6 model to investigate the impact of Greenland Ice Sheet (GrIS) melting, a key uncertainty for past and future AMOC changes. The ocean-only simulation at 1/24° resolution in the Arctic-North Atlantic Ocean performed over the period 2004–2016 indicates that the spread of the Greenland freshwater runoff toward the center of the Labrador Sea, where oceanic convection occurs, seems larger in this model than in a CMIP6 model. Potential explanations are related to the model spatial resolution and the representation of mesoscale processes, which more realistically transport the freshwater released around the shelves and, through eddies, provides strong lateral exchanges between the fine-scale boundary current and the convective basin in the Labrador Sea. The larger freshening of the Labrador Sea in the high-resolution model then strongly affects deep convection activity. In the simulation including GrIS melting, the AMOC weakens by about 2 Sv after only 13 years, far more strongly than what is found in the CMIP6 model. This difference raises serious concerns on the ability of CMIP6 models to correctly assess the potential impact of GrIS melting on the AMOC changes over the last few decades as well as on its future fate. To gain confidence in the GrIS freshwater impacts on climate simulations and therefore in AMOC projections, urgent progress should be made on the parameterization of mesoscale processes in ocean models.