Summary: Missing data is inevitable in many situations that could hamper data analysis for scientific investigations. We establish flexible analytical tools for multivariate skew t models when fat-tailed, asymmetric and missing observations simultaneously occur in the input data. For the ease of computation and theoretical developments, two auxiliary indicator matrices are incorporated into the model for the determination of observed and missing components of each observation that can effectively reduce the computational complexity. Under the missing at random assumption, we present a Monte Carlo version of the expectation conditional maximization algorithm, which is performed to estimate the parameters and retrieve each missing observation with a single value. Additionally, a Metropolis-Hastings within Gibbs sampler with data augmentation is developed to account for the uncertainty of parameters as well as missing outcomes. The methodology is illustrated through two real data sets.