The main challenges of age estimation from facial expression videos lie not only in the modeling of the static facial appearance, but also in the capturing of the temporal facial dynamics. Traditional techniques to this problem focus on constructing handcrafted features to explore the discriminative information contained in facial appearance and dynamics separately. This relies on sophisticated feature-refinement and framework-design. In this paper, we present an end-to-end architecture for age estimation, called Spatially-Indexed Attention Model (SIAM), which is able to simultaneously learn both the appearance and dynamics of age from raw videos of facial expressions. Specifically, we employ convolutional neural networks to extract effective latent appearance representations and feed them into recurrent networks to model the temporal dynamics. More importantly, we propose to leverage attention models for salience detection in both the spatial domain for each single image and the temporal domain for the whole video as well. We design a specific spatially-indexed attention mechanism among the convolutional layers to extract the salient facial regions in each individual image, and a temporal attention layer to assign attention weights to each frame. This two-pronged approach not only improves the performance by allowing the model to focus on informative frames and facial areas, but it also offers an interpretable correspondence between the spatial facial regions as well as temporal frames, and the task of age estimation. We demonstrate the strong performance of our model in experiments on a large, gender-balanced database with 400 subjects with ages spanning from 8 to 76 years. Experiments reveal that our model exhibits significant superiority over the state-of-the-art methods given sufficient training data.