In probability and statistics, uncertainty is usually quantified using single-valued probabilities satisfying Kolmogorov's axioms. Generalisation of classical probability theory leads to various less restrictive representations of uncertainty which are collectively referred to as imprecise probability. Several imprecise approaches to statistical inference using imprecise probability have been suggested, one of which is nonparametric predictive inference (NPI). The multinomial NPI model was recently proposed, which quantifies uncertainty in terms of lower and upper probabilities. It has several advantages, one being the facility to handle multinomial data sets with unknown numbers of possible outcomes. The model gives inferences about a single future observation. This thesis comprises new theoretical developments and applications of the multinomial NPI model. The model is applied to selection problems, for which multiple future observations are also considered. This is the first time inferences about multiple future observations have been presented for the multinomial NPI model. Applications of NPI to classification are also considered and a method is presented for building classification trees using the maximum entropy distribution consistent with the multinomial NPI model. Two algorithms, one approximate and one exact, are proposed for finding this distribution. Finally, a new NPI model is developed for the case of multinomial data with subcategories and several properties of this model are proven.