Abstract. Run-time monitors ensure that untrusted software and system behavior adheres to a security policy. This paper defines an expressive formal framework, based on I/O automata, for modeling systems, policies, and run-time monitors in more detail than is typical. We explicitly model, for example, the environment, applications, and the interaction between them and monitors. The fidelity afforded by this framework allows us to explicitly formulate and study practical constraints on policy enforcement that were often only implicit in previous models, providing a more accurate view of what can be enforced by monitoring in practice. We introduce two definitions of enforcement, target-specific and generalized, that allow us to reason about practical monitoring scenarios. Finally, we provide some meta-theoretical comparison of these definitions and we apply them to investigate policy enforcement in scenarios where the monitor designer has knowledge of the target application and show how this can be exploited to make more efficient design choices.