Fractures are abundant in many geological formations and are often the main pathways for fluid flow. They hence control many different geological processes, ranging from oil production from the world's largest hydrocarbon reservoirs to heat extraction from enhanced geothermal systems, subsurface storage of greenhouse gases, or the migration of methane in gas-bearing sediments - even the formation of many world class ore deposits is, primarily, controlled by the presence of fractures. Yet, it is often the fluid transfer between fractures and matrix, driven by capillary forces, which determines, for example, how well hydrocarbons can be extracted from the subsurface or how readily greenhouse gases are trapped in a geological formation. This talk will discuss how novel pore-scale modelling techniques can be used to analyse the emergent behaviour of capillary forces in complex porous media, how capillary-driven exchange between fractures and matrix can be quantified using a universally applicable scaling law, and how both aspects can be combined to develop more robust and much-needed conceptual models that describe multi-phase flow in fractured geological formations.