"Overlearning" means that a model trained for a seemingly simple objective implicitly learns to recognize attributes and concepts that are (1) not part of the learning objective, and (2) sensitive from a privacy or bias perspective. For example, a binary gender classifier of facial images also learns to recognize races\textemdash even races that are not represented in the training data\textemdash and identities. We demonstrate overlearning in several vision and NLP models and analyze its harmful consequences. First, inference-time representations of an overlearned model reveal sensitive attributes of the input, breaking privacy protections such as model partitioning. Second, an overlearned model can be "re-purposed" for a different, privacy-violating task even in the absence of the original training data. We show that overlearning is intrinsic for some tasks and cannot be prevented by censoring unwanted attributes. Finally, we investigate where, when, and why overlearning happens during model training.