We discuss estimation of the differentiated products demand system of Berry et al (1995) (BLP) by maximum likelihood estimation (MLE). We derive the maximum likelihood estimator in the case where prices are endogenously generated by firms that set prices in Bertrand-Nash equilibrium. In Monte Carlo simulations the MLE estimator outperforms the best-practice GMM estimator on both bias and mean squared error when the model is correctly specified. This remains true under some forms of misspecification. In our simulations, the coverage of the ML estimator is close to its nominal level, whereas the GMM estimator tends to under-cover. We conclude the paper by estimating BLP on the car data used in the original Berry et al (1995) paper, obtaining similar estimates with considerably tighter standard errors.