Today, the most effective mechanism for remedying shortcomings of the Internet, or augmenting it with new networking capabilities, is to develop and deploy a new overlay network. This leads to the problem of multiple networking infrastructures, each with independent advantages, and each developed in isolation. A greatly preferable solution is to have a single infrastructure under which new overlays can be developed, deployed, selected, and combined according to application and administrator needs. MOSAIC is an extensible infrastructure that enables not only the specification of new overlay networks, but also dynamic selection and composition of such overlays. MOSAIC provides declarative networking: it uses a unified declarative language (Mozlog) and runtime system to enable specification of new overlay networks, as well as their composition in both the control and data planes. Importantly, it permits dynamic compositions with both existing overlay networks and legacy applications. This paper demonstrates the dynamic selection and composition capabilities of MOSAIC with a variety of declarative overlays: an indirection overlay that supports mobility (i3), a resilient overlay (RON), and a transport-layer proxy. Using a remarkably concise specification, MOSAIC provides the benefits of runtime composition to simultaneously deliver application-aware mobility, NAT traversal and reliability with low performance overhead, demonstrated with deployment and measurement on both a local cluster and the PlanetLab testbed.