Cavity-mediated cooling has emerged as the only general technique with the potential to cool molecular species down to the microkelvin temperatures needed for quantum coherence and degeneracy. The EuroQUAM CMMC project will link leading theoreticians and experimentalists, including the technique's inventors and experimental pioneers, to develop it into a truly practical technique, reinforcing European leadership in this field. Four major experiments will explore a spectrum of complementary configurations and cavity-mediated cooling will be applied to molecules for the first time; a comprehensive theoretical programme will meanwhile examine the underlying mechanisms and identify the optimal route to practicality. The close connections between theory and experiment, and between pathfinding and underpinning studies, will allow each to guide and inform the others, ensuring that cavity-mediated cooling is swiftly developed as a broad enabling technology for new realms of quantum coherent molecular physics and chemistry.The Southampton component will address, both experimentally and theoretically, fundamental aspects of the cooling process that result from the retarded interaction of a trapped molecule with its reflection in a single mirror, and developments of this prototype scheme that exploit nanostructured mirror arrays that can be produced in our fabrication facilities, and which show both geometric and plasmonic resonances. Our particular aims are hence to understand and explore the most basic version of cavity-mediated cooling, and to develop new implementations suitable for nanoscale integration as a future technology.