This talk will present research that takes place at the University of Manchester on Grid Scheduling. The talk will argue for the need to understand better the mechanics of scheduling on the Grid and heterogeneous systems in general. The argument will be illustrated using essentially three different problems where better understanding and appropriate heuristics for scheduling at the middleware level can improve application performance significantly. The three problems are related to: (i) mapping workflows represented as a Directed Acyclic Graph; (ii) mapping parallel query plans on the Grid; and (iii) scheduling jobs onto compute resources using brokers and service level agreements. Solutions and experimental results to aspects of those problems will also be presented.