While resource management and task scheduling are identified challenges of Grid computing, current Grid scheduling systems mainly focus on CPU and network availability. Recent performance improvement of CPU and computer network has made memory usage a significant factor of overall performance. In this study, we consider memory availability as a performance factor and introduce memory conscious task partition and scheduling. Three task partition policies are discussed. They are CPU-based, memory-based, and CPU-memory combined partition. We first investigate the three task partition policies on dedicated resources and verify the effectiveness of the CPU-memory combined partition algorithm in finding an optimal solution. We then extend the task partition policies in non-dedicated environments with the consideration of resource sharing. Analytical and experimental results show that the CPU-memory combined scheduling approach outperforms either the CPU-based or memory-based scheduling approach considerably for memory-intensive applications in Grid environments.