In many computer vision tasks, the relevant information to solve the problem at hand is mixed to irrelevant, distracting information. This has motivated researchers to design attentional models that can dynamically focus on parts of images or videos that are salient, e.g., by down-weighting irrelevant pixels. In this work, we propose a spatiotemporal attentional model that learns where to look in a video directly from human fixation data. We model visual attention with a mixture of Gaussians at each frame. This distribution is used to express the probability of saliency for each pixel. Time consistency in videos is modeled hierarchically by: 1) deep 3D convolutional features to represent spatial and short-term time relations and 2) a long short-term memory network on top that aggregates the clip-level representation of sequential clips and therefore expands the temporal domain from few frames to seconds. The parameters of the proposed model are optimized via maximum likelihood estimation using human fixations as training data, without knowledge of the action in each video. Our experiments on Hollywood2 show state-of-the-art performance on saliency prediction for video. We also show that our attentional model trained on Hollywood2 generalizes well to UCF101 and it can be leveraged to improve action classification accuracy on both datasets.