In this thesis we present a hierarchical Bayesian methodology for analyzing polychotomous data from multi-stage cluster samples. We begin with a model for multinomial data drawn from a two-stage cluster sample of a finite population. This model is then extended to incorporate partially observed data assuming that the data are missing at random (MAR), in the terminology of Little and Rubin (1987). We next develop a model for polychotomous data collected via a three-stage cluster sample. As with the two-stage model, we describe the methodology for dealing with partially observed data assuming they are MAR. We apply these two methodologies to the 1990 Slovenian Public Opinion Survey and present the results of these analyses. Finally, we fashion a multivariate probit model for a special type of multinomial data, multivariate binary data. We then construct this model that incorporates covariate information for the case of a two-stage cluster sample. Specifically, we outline this methodology for a two-stage cluster sample. This approach also allows for the integration of missing data into the analysis if the data are MAR. For all of the above models we use Markov chain Monte Carlo techniques to simulate samples from the posterior distribution. These samples are then utilized in making inference from the models.