Contract theory is an appealing framework for rigorous component-based design of highly dynamic and distributed cyber-physical systems. Formally, a contract is a specification consisting of pairs of assumptions and guarantees. A guarantee describes the task that the component must fulfill when its environment (made of other components and of the external environment) satisfies the associated assumption. Assume-guarantee contracts make it possible to design components that can adapt under dynamic and uncertain working conditions. Moreover, compositional reasoning makes it possible to prove properties of a system based on the contracts satisfied by its components. In this presentation, we will explore the connections between assume-guarantee contracts and set invariance, a concept which forms one of the cornerstones of modern control theory. We will introduce a class of assume-guarantee contracts whose satisfaction can be characterized through invariant sets of an auxiliary dynamical system. We will discuss theoretical and computational aspects of the proposed framework and show illustrative applications in the design of hierarchical and distributed control systems.