This article summarizes recent advances in the modelling of ordered categorical (ordinal) response variables. We begin by reviewing some models for ordinal data introduced in the literature in the past 25 years. We then survey recent extensions of these models and related methodology for special types of applications, such as for repeated measurement and other forms of clustering. We also survey other aspects of ordinal modelling, such as small-sample analyses, power and sample size considerations, and availability of software. Throughout, we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data.