The numerical methods of conventional Eulerian models have obscured our fundamental understanding of cloud microphysics and introduced artificial uncertainties. In contrast, the Super-Droplet Model (SDM) provides a transparent link between model and theory, and remedies the numerical artifacts that hindered decades of cloud modelling. In light of its numerous advantages we’ve created a novel SDM for warm-cloud microphysics called CLEO, with the goal of making it feasible to run Large-Eddy Simulations (LES) with SDM in domains large enough to resolve shallow mesoscale cloud organisation O(100 km). Here we document the microphysics grounding CLEO and how it is translated into numerical methods, with the intention of assisting the physical interpretation of future LES and comparison with observations. We highlight subtle but important points where we differ from existing SDMs: in how we model the ventilation effect on evaporation, and how we account for uncertainty in our knowledge of droplet collisions. As well as modelling collision-coalescence we propose a low-cost extension to the original SDM algorithm which adds both collisional rebound and breakup. We demonstrate CLEO’s capabilities with known test-cases for condensation/evaporation, collisions between droplets, and droplet motion, including an integrated test using the 1-D Kinematic Driver framework. CLEO can therefore now be used stand-alone, one-way coupled for piggybacking, or two-way coupled for LES, as a fully-functioning SDM capable of representing all the main microphysical processes driving warm-clouds.