The Computational Grid is a promising platform for the efficient execution of parameter sweep applications over large parameter spaces. To achieve performance on the Grid, such applications must be scheduled so that shared data files are strategically placed to maximize reuse, and so that the application execution can adapt to the deliverable performance potential of target heterogeneous, distributed and shared resources. Parameter sweep applications are an important class of applications and would greatly benefit from the development of Grid middleware that embeds a scheduler for performance and targets Grid resources transparently. In this paper we describe a user-level Grid middleware project, the AppLeS Parameter Sweep Template (APST), that uses application-level scheduling techniques  and various Grid technologies to allow the efficient deployment of parameter sweep applications over the Grid. We discuss several possible scheduling algorithms and detail our software design. We then describe our current implementation of APST using systems like Globus , NetSolve  and the Network Weather Service , and present experimental results.