Over 100,000 refugees are permanently resettled from refugee camps to hosting countries every year. Nevertheless, refugee resettlement processes in most hosing countries are ad hoc, accounting for neither the needs and capacities of hosting communities nor the preferences of refugees themselves. Building on models from two-sided matching theory, we introduce a new and very general combinatorially-constrained matching problem to capture the multiunit and multidimensional needs of refugees and capacities of localities. We propose four refugee resettlement mechanisms that can be used by hosting countries under various institutional and informational constraints. These mechanisms can improve match efficiency, give refugees incentives to report where they would best like to resettle, and respect the priorities of localities, thereby encouraging them to accept more refugees in total. Our model is of independent theoretical interest and has many applications in school choice, doctor-hospital matching, teacher allocation, kindergarten slot allocation, and team organization.