A review is given of the studies of the dimensionality characteristics of percolation clusters. The purely geometric nature of a percolation phase transition and the great variety of the quantities exhibiting critical behavior make this geometric approach both informative and useful. In addition to the fractal dimensionality of a cluster and its subsets (such as the backbone, hull, and other dimensionalities), it is necessary to introduce additional characteristics. For example, the maximum velocity of propagation of excitations is determined by the chemical dimensionality of a cluster, and the critical behavior of the conductivity, diffusion coefficient, etc., is determined by spectral (or other related to it) dimensionalities. Scaling relationships between different dimensionalities, as well as relationships between dimensionalities and conventional critical exponents are discussed.