Using grid resources to execute scientific applications requiring a large amount of computing power is attractive but not easy from the user point of view. Vigne is a grid operating system designed to provide a simplified view of a grid to users. Vigne deals with the very large number of nodes in a large-scale grid and with the nodes' dynamic behavior by using peer-to-peer overlays as a keystone. In this paper, we show why it is highly desirable to use structured and unstructured peer-to-peer overlays for building the high-level services of Vigne grid operating system. To show the interest of our approach, we detail the features of two Vigne services built on top of peer-to-peer overlays. We also present experimental results obtained on the Grid'5000 testbed showing the scalability of the Vigne infrastructure based on overlays and its practical interest for the implementation of Vigne distributed services.