In this paper, we review analytic methods for regression models for longitudinal categorical responses. We focus on both likelihood-based approaches and non-likelihood approaches to analysing repeated binary responses. In both approaches, interest is focussed primarily on the regression parameters for the marginal expectations of the binary responses. The association or time dependence between the responses is largely regarded as a nuisance characteristic of the data. We consider these approaches for both the complete and incomplete data cases. We describe the generalized estimating equations (GEE) approach, a non-likelihood approach, and some proposed extensions of it. We also discuss likelihood-based approaches that are based on a log-linear representation of the joint probabilities of the binary responses. We describe how a likelihood-based "mixed parameter" model yields likelihood equations for the regression parameters that are of exactly the same form as the GEE. An outline of the desirable features and drawbacks of each approach is presented. In addition, we provide some comparisons in terms of asymptotic relative efficiency for the complete data case, and in terms of asymptotic bias for the incomplete data case. Finally, we make some recommendations concerning the application of these methods.