The allocation and exchange of discrete resources, such as transplant
organs, public housing, dormitory rooms, and many other resources
for which agents have single-unit demand, is often conducted via direct
mechanisms without monetary transfers. Incentive compatibility and
efficiency are primary concerns when designing such mechanisms. We
construct the full class of group strategy-proof and Pareto efficient
mechanisms and show that each of them can be implemented by endowing
agents with control rights over resources. This new class, which we
call trading cycles, contains new mechanisms as well as known mechanisms
such as top trading cycles, serial dictatorships, and hierarchical
exchange. We illustrate how one can use our construction to show what
can and what cannot be achieved in a variety of allocation and exchange
problems, and we provide an example in which the new trading-cycles
mechanisms are more Lorenz equitable than all previously known mechanisms.