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Fine-tuning has become the standard practice for adapting pre-trained models to downstream tasks. However, the impact on model robustness is not well understood. In this work, we characterize the robustness-accuracy trade-off in fine-tuning. We evaluate the robustness and accuracy of fine-tuned models over 6 benchmark datasets and 7 different fine-tuning strategies. We observe a consistent trade-off between adversarial robustness and accuracy. Peripheral updates such as BitFit are more effective for simple tasks---over 75% above the average measured by the area under the Pareto frontiers on CIFAR-10 and CIFAR-100. In contrast, fine-tuning information-heavy layers, such as attention layers via Compacter, achieves a better Pareto frontier on more complex tasks---57.5% and 34.6% above the average on Caltech-256 and CUB-200, respectively. Lastly, we observe that the robustness of fine-tuning against out-of-distribution data closely tracks accuracy. These insights emphasize the need for robustness-aware fine-tuning to ensure reliable real-world deployments.