In this paper, we address the problem of content placement in peer-to-peer (P2P) systems, with the objective of maximizing the utilization of peers' uplink bandwidth resources. We consider system performance under a many-user asymptotic. We distinguish two scenarios, namely "Distributed Server Networks" (DSNs) for which requests are exogenous to the system, and "Pure P2P Networks" (PP2PNs) for which requests emanate from the peers themselves. For both scenarios, we consider a loss network model of performance and determine asymptotically optimal content placement strategies in the case of a limited content catalog. We then turn to an alternative "large catalog" scaling where the catalog size scales with the peer population. Under this scaling, we establish that storage space per peer must necessarily grow unboundedly if bandwidth utilization is to be maximized. Relating the system performance to properties of a specific random graph model, we then identify a content placement strategy and a request acceptance policy that jointly maximize bandwidth utilization, provided storage space per peer grows unboundedly, although arbitrarily slowly, with system size.