The Grid vision is to allow heterogeneous computational resources to be shared and utilised globally. Grid users are able to submit tasks to remote resources for execution. However, these resources may be unreliable and there is a risk that submitted tasks may fail or cost more than expected. The notion of trust is often used in agent-based systems to manage such risk, and in this paper we apply trust to the problem of resource selection in Grid computing. We propose a number of resource selection algorithms based upon trust, and evaluate their effectiveness in a simulated Grid.