We study the allocation and exchange of discrete resources--such as kidneys or school seats--in environments in which monetary transfers are not allowed. We introduce a new class of direct mechanisms that we call trading cycles, and show that a mechanism is group dominant-strategy incentive compatible and Pareto efficient if and only if it is equivalent to a trading-cycles mechanism. The class contains new mechanisms as well as such previously studied mechanisms as top trading cycles, serial dictatorships, and hierarchical exchange. In some problems, the new trading-cycles mechanisms perform better than all previously known mechanisms. Just as importantly, knowing that all group incentive-compatible and efficient mechanisms belong to our class allows us to easily determine which efficient outcomes can and cannot be achieved in a group incentive-compatible way.