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State-of-the-art object recognition methods do not generalize well to unseen domains. Work in domain generalization has attempted to bridge domains by increasing feature compatibility, but has focused on standard, appearance-based representations. We show the potential of shape-based representations to increase domain robustness. We compare two types of shape-based representations: one trains a convolutional network over edge features, and another computes a soft, dense medial axis transform. We show the complementary strengths of these representations for different types of domains, and the effect of the amount of texture that is preserved. We show that our shape-based techniques better leverage data augmentations for domain generalization, and are more effective at texture bias mitigation than shape-inducing augmentations. Finally, we show that when the convolutional network in state-of-the-art domain generalization methods is replaced with one that explicitly captures shape, we obtain improved results.