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Real-world object detection models should be cheap and accurate. Knowledge distillation (KD) can boost the accuracy of a small, cheap detection model by leveraging useful information from a larger teacher model. However, a key challenge is identifying the most informative features produced by the teacher for distillation. In this work, we show that only a very small fraction of features within a ground-truth bounding box are responsible for a teacher's high detection performance. Based on this, we propose Prediction-Guided Distillation (PGD), which focuses distillation on these key predictive regions of the teacher and yields considerable gains in performance over many existing KD baselines. In addition, we propose an adaptive weighting scheme over the key regions to smooth out their influence and achieve even better performance. Our proposed approach outperforms current state-of-the-art KD baselines on a variety of advanced one-stage detection architectures. Specifically, on the COCO dataset, our method achieves between +3.1% and +4.6% AP improvement using ResNet-101 and ResNet-50 as the teacher and student backbones, respectively. On the CrowdHuman dataset, we achieve +3.2% and +2.0% improvements in MR and AP, also using these backbones. Our code is available at https://github.com/ChenhongyiYang/PGD.