This paper focuses on kidney exchange in a setting with non-dichotomous preferences motivated by the use of desensitization. A planner in charge of a kidney exchange program can improve the welfare of patients by sorting them into “priority groups”. Priority may be given to, e.g., patients that are difficult to find compatible donors for or patients with severe conditions. The planner may impose any restrictions on the types of exchanges that are permitted. Priority group matchings are a new class of Pareto efficient matchings that take priority groups into account. Several well-known classes of matchings are shown to be special cases of Priority group matchings, corresponding to different approaches to prioritizing patients and different institutional designs. Priority group matchings are characterized in terms of solutions to an integer programming problem. When preferences are non-dichotomous, Pareto efficiency is no longer equivalent to maximality. Two characterizations of Pareto efficient matchings are provided in this setting.