Peer-to-peer overlays allow distributed applications to work in a wide-area, scalable, and fault-tolerant manner. However, most structured and unstructured overlays present in literature today are inflexible from the application viewpoint. In other words, the application has no control over the structure of the overlay itself. This paper proposes the concept of an application-malleable overlay, and the design of the first malleable overlay which we call MOve. In MOve, the communication characteristics of the distributed application using the overlay can influence the overlay's structure itself, with the twin goals of (1) optimizing the application performance by adapting the overlay, while also (2) retaining the large scale and fault tolerance of the overlay approach. The influence could either be explicitly specified by the application or implicitly gleaned by our algorithms. Besides neighbor list membership management, MOve also contains algorithms for resource discovery, update propagation, and churn-resistance. The emergent behavior of the implicit mechanisms used in MOve manifests in the following way: when application communication is low, most overlay links keep their default configuration; however, as application communication characteristics become more evident, the overlay gracefully adapts itself to the application.