Methods of Rubin (1983) for robust estimation of a mean and covariance matrix and associated parameters are extended to analyse data with missing values. The methods are maximum likelihood ($ML$) for multivariate $t$ and contaminated normal models. $ML$ estimation is achieved by the $EM$ algorithm, and involves minor modifications to the $EM$ algorithm for multivariate normal data. The methods are shown to be superior to existing methods in a simulation study, using data generated from a variety of models. Model selection and standard error estimation are discussed with the aid of two real data examples.