The neural plausibility of backpropagation has long been disputed, primarily for its use of instantaneous weight transport. A variety of prior proposals that avoid weight transport fail to scale on complex tasks such as ImageNet; however, a recent proposal has reported competitive performance with backpropagation. We find that this local learning rule requires complex hyperparameter tuning that does not transfer across architectures. We identify a more robust local learning rule that transfers to deep networks on ImageNet without re-tuning hyperparameters. Nonetheless, we find a performance gap between this local rule and backpropagation that widens with increasing model depth. We formulate several non-local learning rules that address these shortcomings without requiring strict weight symmetry and match state-of-the-art performance on networks with hundreds of layers, even in the presence of noisy updates. Our results suggest two routes towards the discovery of neural implementations for credit assignment: further improvement of local rules so that they perform consistently across architectures and the identification of scalable biological implementations for our non-local learning mechanisms.