Design of matching systems between refugees and states or local areas is emerging as one of the most promising solutions to problems in refugee resettlement. We describe the basics of two-sided matching theory used in a number of allocation problems, such as school choice, where both sides need to agree to the match. We then explain how these insights can be applied to international refugee matching in the context of the European Union and examine how refugee matching might work within the United Kingdom, Canada, and the United States.