We present the mathematical foundations of the contract-based model developed in the framework of the SPEEDS project. SPEEDS aims at developing methods and tools to support "speculative design", a design methodology in which distributed designers develop different aspects of the overall system, in a concurrent but controlled way. Our generic mathematical model of contract supports this style of development. This is achieved by focusing on behaviors, by supporting the notion of "rich component" where diverse (functional and non-functional) aspects of the system can be considered and combined, by representing rich components via their set of associated contracts, and by formalizing the whole process of component composition.