We address the problem of detecting attention targets in video. Our goal is to identify where each person in each frame of a video is looking, and correctly handle the case where the gaze target is out-of-frame. Our novel architecture models the dynamic interaction between the scene and head features and infers time-varying attention targets. We introduce a new annotated dataset, VideoAttentionTarget, containing complex and dynamic patterns of real-world gaze behavior. Our experiments show that our model can effectively infer dynamic attention in videos. In addition, we apply our predicted attention maps to two social gaze behavior recognition tasks, and show that the resulting classifiers significantly outperform existing methods. We achieve state-of-the-art performance on three datasets: GazeFollow (static images), VideoAttentionTarget (videos), and VideoCoAtt (videos), and obtain the first results for automatically classifying clinically-relevant gaze behavior without wearable cameras or eye trackers.