A distributed system is no longer confined to a single administrative domain. Peer-to-peer applications and business-to-business e-commerce systems, for example, typically span multiple local-area and wide-area networks, raising issues of trust, security, and anonymity. This paper introduces a distributed systems model with an explicit notion of domain that defines the scope of trust and local communication within a system. We introduce leader-election oracles that distinguish between common and distinct domains, encapsulating failure-detection information and leading to modular solutions and proofs. We show how Reliable Broadcast can be implemented in our domain-based model, we analyze the cost of communicating across groups, and we establish lower-bounds on the number of cross-domain messages necessary to implement Reliable Broadcast.