Regularization techniques such as the lasso (Tibshirani 1996) and elastic net (Zou and Hastie 2005) can be used to improve regression model coefficient estimation and prediction accuracy, as well as to perform variable selection. Ordinal regression models are widely used in applications where the use of regularization could be beneficial; however, these models are not included in many popular software packages for regularized regression. We propose a coordinate descent algorithm to fit a broad class of ordinal regression models with an elastic net penalty. Furthermore, we demonstrate that each model in this class generalizes to a more flexible form, that can be used to model either ordered or unordered categorical response data. We call this the (ELMO) class, and it includes widely used models such as multinomial logistic regression (which also has an ordinal form) and ordinal logistic regression (which also has an unordered multinomial form). We introduce an elastic net penalty class that applies to either model form, and additionally, this penalty can be used to shrink a non-ordinal model toward its ordinal counterpart. Finally, we introduce the R package , which implements the algorithm for this model class.