Properties of the coincidence set of two mappings are studied. Both single-valued and set-valued mappings are considered. Estimates for the cardinality of the coincidence set are obtained for mappings of metric and partially ordered spaces. For mappings of a normed space to a metric space necessary and sufficient conditions that there exist at least two coincidence points, sufficient conditions that there exist at least n coincidence points, and sufficient conditions that the coincidence set is infinite are given. For abstract inclusions in metric and normed spaces necessary and sufficient conditions that at least one solution exists, sufficient conditions that there exist at least n solutions, and sufficient conditions that the solution set is infinite are put forward. All the results obtained are equally meaningful for set-valued and single-valued mappings.