Specialized accelerators such as GPUs, TPUs, FPGAs, and custom ASICs have been increasingly deployed to train deep learning models. These accelerators exhibit heterogeneous performance behavior across model architectures. Existing schedulers for clusters of accelerators, which are used to arbitrate these expensive training resources across many users, have shown how to optimize for various multi-job, multi-user objectives, like fairness and makespan. Unfortunately, existing schedulers largely do not consider performance heterogeneity. In this paper, we propose Gavel, a heterogeneity-aware scheduler that systematically generalizes a wide range of existing scheduling policies. Gavel expresses these policies as optimization problems, making it easy to optimize for objectives in a heterogeneity-aware way, while also being cognizant of performance optimizations like space sharing. Gavel then uses a round-based scheduling mechanism to ensure jobs receive their ideal allocation given the target scheduling policy. Gavel's heterogeneity-aware policies allow a heterogeneous cluster to sustain higher input load, and improve end objectives such as average job completion time and makespan by up to 3.5x compared to heterogeneity-agnostic policies.