Abstract A survey is given of recent results on fine zonotopal tilings by cubes (briefly, cubillages) of cyclic zonotopes. The main interest of this theory is that it is interrelated with the theory of higher Bruhat orders, as well as with the parallel theory of triangulations of cyclic polytopes and Tamari–Stasheff posets, used in investigations of the Kadomtsev– Petviashvili equations and higher Auslander–Reiten algebras. Bibliography: 35 titles.