Inference for survey data needs to take account of the survey design. Failing to consider the survey design in inference may lead to misleading results. The standard analysis of categorical data, developed under the assumption of multinomial sampling, is inadequate as the commonly used sampling schemes clearly violate this assumption. Since, Kish (1965) introduced the idea of a design effect, many classical solutions have been proposed, such as, first- and second-order corrections to Pearson chi-squared, likelihood-ratio chi-squared, and Wald tests. Our objective in this thesis is to present an investigation of a Bayesian approach to the analysis of categorical survey data, arising from designs including simple random sampling, finite population sampling, stratification, and cluster sampling. We focus on Bayesian methods for model selection and model averaging, where Bayes factors and the Bayesian Information Criterion (BIC) approximation have been offered as alternative approaches. These Bayesian methods are reviewed, and comparisons made between their performance. The effect of ignoring the complex sampling design is investigated. Moreover, adjustments to the multinomial-based Bayes factor and BIC are produced and evaluated.