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Thermal hot spots and high temperature gradients degrade reliability and performance, and increase cooling costs and leakage power. In this paper, we explore the benefits of temperature-aware task scheduling for multiprocessor system-on-a-chip (MPSoC). We evaluate our techniques using workload characteristics collected from a real system by Sun's Continuous System Telemetry. We first solve the task scheduling problem statically using integer linear programming (ILP). The ILP solution is guaranteed to be optimal for the given assumptions for tasks. We formulate ILPs for minimizing energy, balancing energy, and reducing hot spots, and provide an extensive comparison of their thermal behavior against our technique. Our static solution can reduce the frequency of hot spots by 35%, spatial gradients by 85%, and thermal cycles by 61% in comparison to the ILP for minimizing energy. We then design dynamic scheduling policies at the OS-level with negligible performance overhead. Our adaptive dynamic policy reduces the frequency of high-magnitude thermal cycles and spatial gradients by around 50% and 90%, respectively, in comparison to state-of-the-art schedulers. Reactive thermal management strategies, such as thread migration, can be combined with our scheduling policy to further reduce hot spots, temperature variations, and the associated performance cost.