Growth mixture models (GMMs) with non-ignorable missing data have drawn increasing attention in research communities, but few studies have discussed robust growth mixture models with non-ignorable missing data. Blindly using normal theory based models for non-normal data may result in biased point estimators, stand errors, and associated confidence intervals. Also, ignoring the non-ignorable missingness may pose substantial risk of reaching incorrect conclusions. This dissertation proposes and evaluates a Bayesian approach to robust growth mixture models with non-ignorable missing data. First, backgrounds are introduced on missing data, robust methods, and the normal approach to estimate growth mixture models with missing data through Bayesian methods. Second, robust growth models with non-ignorable missing are proposed. Three growth models are reviewed; Three robust growth models are presented, in which measurement errors and random effects are assumed to follow t distributions for heavy-tailed data. Four non-ignorable missingness selection models are formulated with robust growth models. Third, a full Bayesian method is then proposed to estimate the models. Through the data augmentation algorithm, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. Fourth, model selection criteria in the Bayesian context are proposed. Fifth, 5 simulation studies are conducted. Results from different models with different data distributions, missingness mechanisms, sample sizes, class separation (or distance), number of classes are analyzed and compared. The main conclusions drawn from simulation studies are (1) the proposed Bayesian method recovers model parameters very well; (2) almost all the proposed model selection criteria can identify true models with very high certainties; and (3) ignoring the non-ignorable missingness or incorrectly modeling data distribution will cause misleading conclusions. Suggestions for real data analysis in practice are provided. Sixth, a real data set of mathematical ability is analyzed to demonstrate the application of the model and the method. Last, the model selection criteria, the convergence problem, the sample size, the sensitivity of the selection model, the number of classes, the missingness, the number of replications, and the future directions of the approach are discussed.