Accurately representing coastal and shelf seas in global ocean models represents one of the grand challenges of Earth system science. They are regions of immense societal importance through the goods and services they provide, hazards they pose and their role in global-scale processes and cycles, e.g. carbon fluxes and dense water formation. However, they are poorly represented in the current generation of global ocean models. In this contribution, we aim to briefly characterise the problem, and then to identify the important physical processes, and their scales, needed to address this issue in the context of the options available to resolve these scales globally and the evolving computational landscape.We find barotropic and topographic scales are well resolved by the current state-of-the-art model resolutions, e.g. nominal 1∕12°, and still reasonably well resolved at 1∕4°; here, the focus is on process representation. We identify tides, vertical coordinates, river inflows and mixing schemes as four areas where modelling approaches can readily be transferred from regional to global modelling with substantial benefit. In terms of finer-scale processes, we find that a 1∕12° global model resolves the first baroclinic Rossby radius for only ∼ 8% of regions 〈 500m deep, but this increases to ∼ 70% for a 1∕72° model, so resolving scales globally requires substantially finer resolution than the current state of the art.We quantify the benefit of improved resolution and process representation using 1∕12° global- and basin-scale northern North Atlantic nucleus for a European model of the ocean (NEMO) simulations; the latter includes tides and a k-ε vertical mixing scheme. These are compared with global stratification observations and 19 models from CMIP5. In terms of correlation and basin-wide rms error, the high-resolution models outperform all these CMIP5 models. The model with tides shows improved seasonal cycles compared to the high-resolution model without tides. The benefits of resolution are particularly apparent in eastern boundary upwelling zones.To explore the balance between the size of a globally refined model and that of multiscale modelling options (e.g. finite element, finite volume or a two-way nesting approach), we consider a simple scale analysis and a conceptual grid refining approach. We put this analysis in the context of evolving computer systems, discussing model turnaround time, scalability and resource costs. Using a simple cost model compared to a reference configuration (taken to be a 1∕4° global model in 2011) and the increasing performance of the UK Research Councils' computer facility, we estimate an unstructured mesh multiscale approach, resolving process scales down to 1.5km, would use a comparable share of the computer resource by 2021, the two-way nested multiscale approach by 2022, and a 1∕72° global model by 2026. However, we also note that a 1∕12° global model would not have a comparable computational cost to a 1° global model in 2017 until 2027. Hence, we conclude that for computationally expensive models (e.g. for oceanographic research or operational oceanography), resolving scales to ∼ 1.5km would be routinely practical in about a decade given substantial effort on numerical and computational development. For complex Earth system models, this extends to about 2 decades, suggesting the focus here needs to be on improved process parameterisation to meet these challenges.