Methods for estimating probabilities on sample spaces for ordered-categorical variables are surveyed. The methods all involve smoothing the relative frequencies in manners which recognise the ordering among categories. Approaches of this type include convex smoothing, weighting-function and kernel-based methods, near neighbour methods, Bayes-based methods and penalized minimum-distance methods. The relationships among the methods are brought out, application is made to a medical example and a simulation study is reported which compares the methods on univariate and bivariate examples. Links with smoothing procedures in other contexts are indicated.