While deep learning has resulted in major breakthroughs in many application domains, the frameworks commonly used in deep learning remain fragile to artificially-crafted and imperceptible changes in the data. In response to this fragility, adversarial training has emerged as a principled approach for enhancing the robustness of deep learning with respect to norm-bounded perturbations. However, there are other sources of fragility for deep learning that are arguably more common and less thoroughly studied. Indeed, natural variation such as lighting or weather conditions can significantly degrade the accuracy of trained neural networks, proving that such natural variation presents a significant challenge for deep learning. In this paper, we propose a paradigm shift from perturbation-based adversarial robustness toward model-based robust deep learning. Our objective is to provide general training algorithms that can be used to train deep neural networks to be robust against natural variation in data. Critical to our paradigm is first obtaining a model of natural variation which can be used to vary data over a range of natural conditions. Such models may be either known a priori or else learned from data. In the latter case, we show that deep generative models can be used to learn models of natural variation that are consistent with realistic conditions. We then exploit such models in three novel model-based robust training algorithms in order to enhance the robustness of deep learning with respect to the given model. Our extensive experiments show that across a variety of naturally-occurring conditions and across various datasets, deep neural networks trained with our model-based algorithms significantly outperform both standard deep learning algorithms as well as norm-bounded robust deep learning algorithms.