Humans typically move their eyes in "scanpaths" of fixations linked by saccades. Here we present DeepGaze III, a new model that predicts the spatial location of consecutive fixations in a free-viewing scanpath over static images. DeepGaze III is a deep learning-based model that combines image information with information about the previous fixation history to predict where a participant might fixate next. As a high-capacity and flexible model, DeepGaze III captures many relevant patterns in the human scanpath data, setting a new state of the art in the MIT300 dataset and thereby providing insight into how much information in scanpaths across observers exists in the first place. We use this insight to assess the importance of mechanisms implemented in simpler, interpretable models for fixation selection. Due to its architecture, DeepGaze III allows us to disentangle several factors that play an important role in fixation selection, such as the interplay of scene content and scanpath history. The modular nature of DeepGaze III allows us to conduct ablation studies, which show that scene content has a stronger effect on fixation selection than previous scanpath history in our main dataset. In addition, we can use the model to identify scenes for which the relative importance of these sources of information differs most. These data-driven insights would be difficult to accomplish with simpler models that do not have the computational capacity to capture such patterns, demonstrating an example of how deep learning advances can be used to contribute to scientific understanding.