Capacity is often studied under constraints on the channel input signals. This paper investigates the behavior of capacity when constraints are placed on the! channel output signal (as well as generalizations thereof). While such a change in perspective leaves the point-to-point problem (essentially) unchanged, the main conclusion is that in certain network scenarios, including multiple-access and relay situations, tooth the structure of the problem and the conclusions change. For example, capacity results are found for the many-user Gaussian multiple-access channel (MAC) with arbitrarily dependent sources, cooperation, or feedback, and for the nondegraded Gaussian relay network. The investigations are motivated by recent questions arising in spectrum sharing and dynamic spectrum allocation: Multiple independent networks share the same frequency band, but are spatially mostly disjoint. One approach to grant coexistence is via spatial interference power restrictions, imposed at the network level, rather than at the device level. The corresponding capacity question is posed and partially answered in this paper.