We consider the problem of handling missing data with deep latent variable models (DLVMs). First, we present a simple technique to train DLVMs when the training set contains missing-at-random data. Our approach, called MIWAE, is based on the importance-weighted autoencoder (IWAE), and maximises a potentially tight lower bound of the log-likelihood of the observed data. Compared to the original IWAE, our algorithm does not induce any additional computational overhead due to the missing data. We also develop Monte Carlo techniques for single and multiple imputation using a DLVM trained on an incomplete data set. We illustrate our approach by training a convolutional DLVM on incomplete static binarisations of MNIST. Moreover, on various continuous data sets, we show that MIWAE provides extremely accurate single imputations, and is highly competitive with state-of-the-art methods.