This work presents a model that allows the execution of parallel applications in a Grid environment. Our main focus is on how to share the idle cycles of clusters and computers to execute real parallel applications. We present a new model which introduces the notions of locality and adaptability. The locality is used for job allocation, and for job migration. The adaptability provides a simple mechanism to allow clusters to join or leave a Grid. We also propose the middleware architecture to implement the model, and provide some simulation results to show the main characteristics of the model. Copyright © 2004 John Wiley & Sons, Ltd.