The latest assessment report from the Intergovernmental Panel on Climate Change concluded that the Atlantic Meridional Overturning Circulation (AMOC) was very likely to decline over the twenty-first century under all emissions scenarios; however, there was low confidence in the magnitude of the decline. Recent research has highlighted that model biases in the mean climate state can affect the AMOC in its mean state, variability and its response to climate change. Hence, understanding and reducing these model biases is critical for reducing uncertainty in the future changes of the AMOC and in its impacts on the wider climate. We discuss how model biases, in particular salinity biases, influence the AMOC and deep convection. We then focus on biases in the UK HadGEM3-GC3-1 climate model and how these biases change with resolution. We also discuss ongoing model development activities that affect these biases, and highlight priorities for improved representation of processes, such as the position of the North Atlantic Current, transports in narrow boundary current, resolution (or improved parameterization) of eddies and spurious numerical mixing in overflows. This article is part of a discussion meeting issue ‘Atlantic overturning: new observations and challenges’.