This paper integrates deep neural networks (DNNs) into structural economic models to increase flexibility and capture rich heterogeneity while preserving interpretability. Economic structure and machine learning are complements in empirical modeling, not substitutes: DNNs provide the capacity to learn complex, non-linear heterogeneity patterns, while the structural model ensures the estimates remain interpretable and suitable for decision making and policy analysis. We start with a standard parametric structural model and then enrich its parameters into fully flexible functions of observables, which are estimated using a particular DNN architecture whose structure reflects the economic model. We illustrate our framework by studying demand estimation in consumer choice. We show that by enriching a standard demand model we can capture rich heterogeneity, and further, exploit this heterogeneity to create a personalized pricing strategy. This type of optimization is not possible without economic structure, but cannot be heterogeneous without machine learning. Finally, we provide theoretical justification of each step in our proposed methodology. We first establish non-asymptotic bounds and convergence rates of our structural deep learning approach. Next, a novel and quite general influence function calculation allows for feasible inference via double machine learning in a wide variety of contexts. These results may be of interest in many other contexts, as they generalize prior work.