Peer-To-Peer systems are driving a major paradigm shift in the era of genuinely distributed computing. Gnutella is a good example of a Peer-To-Peer success story: a rather simple software enables Internet users to freely exchange files, such as MP3 music files. But it shows up also some of the limitations of current P2P information systems with respect to their ability to manage data efficiently. In this paper we introduce P-Grid, a scalable access structure that is specifically designed for Peer-To-Peer information systems. P-Grids are constructed and maintained by using randomized algorithms strictly based on local interactions, provide reliable data access even with unreliable peers, and scale gracefully both in storage and communication cost. Keywords: Peer-To-Peer computing, Distributed Indexing, Distributed Databases, Randomized Algorithms.