Of a special interest are tilings in hyperbolic n–space Λn. It is natural to extend the study of tiling problems to the hyperbolic plane as well as hyperbolic spaces of higher dimension. In 1974, K. B¨or¨oczky published a construction of tilings of hyperbolic plane Λ2 by a single prototile. In plain words, the construction goes as follows. Let us describe it for 3-space Λ3 first. Consider a collection of concentric horospheres, where consecutive horospheres have equal distance. Each horosphere is conformal to the Euclidean plane R2. So consider a partition of each horosphere into the canonical tiling of R2 by unit squares. Erect on each square a prism, such that the top of the prism is made of four squares of the next layer. This yields a tiling of Λ3, where each tile carries four tiles on its top. These “polyhedral layers” fit together and produce the B¨or¨oczky tiling of the whole hyperbolic 3-space. This construction can be extended to any dimension, yielding tilings of hyperbolic n–space Λn. To obtain corresponding non face-to-face tiling of 3-space Λ3 into convex prismatic equal hexa-faceted polyhedra it is enough every nine-faceted polyhedra of B¨or¨oczky’s tiling to cut into four prismatic polyhedra its coordinate planes of symmetry. The tilings (face-toface and non-face-to-face) of n–dimensional hyperbolic space Λn are under construction almost literally in the same way through partition of corresponding (n − 1)–horospheres into geodesic (n − 1)–cube (cubiliaj). Theorem 1. In the hyperbolic 3–space Λ3, there exists a nonregular non face-to-face tiling (non-normal) composed of congruent convex polyhedral tiles, which can’t be transformed into regular tiling using any permutation of the polyhedral tiles. Our proof makes use of the so-called B¨or¨oczky tiling of hyperbolic Λ3 by congruent polyhedra. An analogous construction works for arbitrary dimension. Theorem 2. In the hyperbolic n–space Λn, there exists a nonregular non face-to-face (non-normal) tiling composed of congruent convex polyhedral tiles, which can’t be transformed into regular tiling using any permutation of the polyhedral tiles. The proposed construction can be considered and as the constructive proof of the theorem of the existence of non-face-to-face tilings in the n – dimensional hyperbolic space into equal, convex and compact polyhedra. The work outlined some possible generalizations of Boroczky’s construction, which in most cases, also allow to construct and non-face-to-face tilings. Features of tilings can constructively prove some general statements concerning, for example, point Delone Sets and Delone tilings. In the article it is also discussed the question of the number of hyperfaces for hyperbolic n–dimensional tile.