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In recent years, experiments have shown that twisted bilayer graphene and other so-called ``moiré materials'' realize a variety of important strongly-correlated electronic phases, such as superconductivity and fractional quantum anomalous Hall states. I will present a rigorous multiple-scales analysis justifying the (single-particle) Bistritzer-MacDonald model, which played a critical role in the prediction of these phases in twisted bilayer graphene. The significance of this model is that it has moiré-periodic coefficients even when the underlying material is aperiodic at the atomic scale, allowing moiré materials to be studied via Floquet-Bloch band theory. I will then discuss generalizations of this model and other mathematical questions related to moiré materials.