Recent work quantifying postural dynamics has attempted to define the repertoire of behaviors performed by an animal. However, a major drawback to these techniques has been their reliance on dimensionality reduction of images which destroys information about which parts of the body are used in each behavior. To address this issue, we introduce a deep learning-based method for pose estimation, LEAP (LEAP Estimates Animal Pose). LEAP automatically predicts the positions of animal body parts using a deep convolutional neural network with as little as 10 frames of labeled data for training. This framework consists of a graphical interface for interactive labeling of body parts and software for training the network and fast prediction on new data (1 hr to train, 185 Hz predictions). We validate LEAP using videos of freely behaving fruit flies (Drosophila melanogaster) and track 32 distinct points on the body to fully describe the pose of the head, body, wings, and legs with an error rate of <3% of the animal's body length. We recapitulate a number of reported findings on insect gait dynamics and show LEAP's applicability as the first step in unsupervised behavioral classification. Finally, we extend the method to more challenging imaging situations (pairs of flies moving on a mesh-like background) and movies from freely moving mice (Mus musculus) where we track the full conformation of the head, body, and limbs.