Despite broad applications of growth curve models, few studies have dealt with a practical issue -- nonnormality of data. Previous studies have used Student's "t" distributions to remedy the nonnormal problems. In this study, robust distributional growth curve models are proposed from a semiparametric Bayesian perspective, in which intraindividual measurement errors follow unknown random distributions with Dirichlet process mixture priors. Based on Monte Carlo simulations, we evaluate the performance of the robust semiparametric Bayesian method and compare it to the robust method using Student's "t" distributions as well as the traditional normal-based method. We conclude that the semiparametric Bayesian method is more robust against nonnormal data. An example about the development of mathematical abilities is provided to illustrate the application of robust growth curve modeling, using school children's Peabody Individual Achievement Test mathematical test scores from the National Longitudinal Survey of Youth 1997 Cohort. [This paper was published in "Structural Equation Modeling" v27 n4 p544-560 2020.]