Cosmic dust grains are tiny solid particles in the interstellar medium of galaxies. Characterising their physical properties is crucial to understand the evolution of galaxies. In the past decade, our knowledge on cosmic dust has increased substantially, thanks to observational data from UV to far-infrared wavelengths. However, one important part of the spectrum, the mm range, has largely remained terra incognita. Until now, the lack of high-quality mm observations, as well as limitations in modelling the interplay between dust and starlight, have prevented us from mapping all the dust in nearby galaxies and firmly characterising its physical properties. We want to exploit new observational data in the mm range and new advances in modelling techniques to accurately model, for the first time, the interplay between starlight and dust in a sizeable sample of nearby spiral galaxies. We will use our models (1) to search for cold dust, (2) to quantify the importance of different emission mechanisms in the mm range, (3) to look for evidence for a variation of the dust properties between and within galaxies, (4) to quantify the contribution of different dust heating mechanisms, and (5) to constrain the dust grain shape distribution in galaxies. We believe that this project can have a wide impact, from detailed studies of nearby galaxies to the understanding of galaxies at high redshift.