Moment methods for analysing repeated binary responses have been proposed by Liang and Zeger, and extended by Prentice and Zhao and Prentice. In these estimating equations, models are proposed for the correlation between the repeated binary responses. We extend Liang and Zeger's method to models for the correlation between repeated nominal or ordinal categorical responses; in particular, when the repeated responses are binary, our methods reduce to Liang and Zeger's method. Our method is illustrated with two datasets. One dataset contains repeated observations of self-assessment of arthritis, an ordered variable with three categories, collected during a randomized comparative study of alternative treatments of patients with rheumatoid arthritis. The second dataset is a longitudinal study of the health effects of air pollution, in which the repeated ordered multinomial response is the wheezing status (no wheeze, wheeze with cold, wheeze apart from cold) of a child at ages 9, 10, 11 and 12 years.