Inter-language interoperability is big business, as the success of Microsoft's .NET and COM and Sun's JVM show. Programming language designers are designing programming languages that reflect that fact - - SML#, Mondrian, and Scala, to name just a few examples, all treat interoperability with other languages as a central design feature. Still, current multi-language research tends not to focus on the semantics of interoperation features, but only on how to implement them efficiently. In this paper, we take first steps toward higher-level models of interoperating systems. Our technique abstracts away the low-level details of interoperability like garbage collection and representation coherence, and lets us focus on semantic properties like type-safety and observable equivalence.Beyond giving simple expressive models that are natural compositions of single-language models, our studies have uncovered several interesting facts about interoperability. For example, higher-order contracts naturally emerge as the glue to ensure that interoperating languages respect each other's type systems. While we present our results in an abstract setting, they shed light on real multi-language systems and tools such as the JNI, SWIG, and Haskell's stable pointers.