The deployment of increasingly large and capable FPGAs has motivated mechanisms for sharing them, but system support for FPGAs is not yet mature. Traditional scheduling algorithms do not account for the unique characteristics of FPGAs, leading to infeasible or inefficient allocations. We propose a novel scheduling policy, called Spatiotemporal FPGA Scheduling, that overcomes these challenges to achieve long-term target allocations by tracking and correcting deviations from targets across management time periods. Compared to traditional algorithms, Spatiotemporal FPGA Scheduling produces allocations that are up to 32% closer to targets, improves average throughput by up to 44%, and improves average FPGA utilization by up to 23%.