As most of the classifications man has constructed during a long period going from Plato and Aristotle to Linnaeus (and even to the XIXth century) were hierarchical classifications, we devote this complete chapter to the exposition of mathematical models of hierarchies. After some historical views on the subject (Sect. 3.2), we introduce (Sect. 3.3) the basic notions of partition, partition lattice, and chain of partitions, this last one being the exact mathematical model of a hierarchical classification, classically represented by a tree diagram. In Sect. 3.4, we give the structure of the set of all hierarchical classifications on a finite set (which allows us to know exhaustively all the possible hierarchies we can make). Then, in Sect. 3.5, we present the exact correspondence between tree diagram, hierarchical structure, and the distance we can define on it, which is an ultrametric. These ideal models and their algebraic representations (Sect. 3.6) are those that the taxonomists want generally to obtain, but the appearance of the real world, in general, is quite far from such nice orders. So, we explain (Sect. 3.7) how we can replace the empirical quasi-chaotic data with due mathematical taxonomies. Though, in many cases, we are able to do that and can easily adapt our numerical models to empirical data, some problems arise in this operation (Sect. 3.8): either because of the existence of mathematical limits within the models (intrinsic instability) or because of the presence of changes in human perception of the world in the course of time (extrinsic instability). However (Sect. 3.9), we list finally some possible answers to these important questions.