The standard class of adversaries considered in cryptography is that of strict polynomial-time probabilistic machines (or circuits). However, expected polynomial-time machines are often also considered. For example, there are many zero-knowledge protocols for which the only simulation techniques known run in expected (and not strict) polynomial-time. In addition, it has been shown that expected polynomial-time simulation is essential for achieving constant-round black-box zero-knowledge protocols. This reliance on expected polynomial-time simulation introduces a number of conceptual and technical difficulties. In this paper, we develop techniques for dealing with expected polynomial-time adversaries in the context of simulation-based security proofs.