Principled methods for analyzing data with missing values, most notably multiple imputation, have become increasingly popular among political scientists. However, existing multiple imputation strategies can struggle to handle the kinds of large and complex datasets that are also becoming common in the discipline. We propose an accurate, fast, and scalable approach to multiple imputation, which we call MIDAS (Multiple Imputation with Denoising Autoencoders). MIDAS employs a class of dimensionality-reducing neural networks known as denoising autoencoders, which corrupt a subset of input data and attempt to reconstruct it through a series of nested nonlinear transformations. We repurpose denoising autoencoders for multiple imputation by treating missing values as an additional portion of corrupted data, drawing imputations from a model trained to minimize the reconstruction error on the originally observed portion. A host of tests involving both real and simulated data illustrate MIDAS's accuracy and scalability. We provide open-source software for implementing MIDAS.