I have written two books: “Properties of Böröczky’s construction in high-dimensional hyperbolic spaces” and “The behavior of geodesics in two-dimensional hyperbolic manifolds” and this two books have been published. Of a special interest are tilings in hyperbolic n–space Λn . 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 . So consider a partition of each horosphere into the canonical tiling of 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 Böröczky tiling of the whole hyperbolic 3-space. This construction can be extended to any dimension, yielding tilings of hyperbolic n–space Λn . The main purpose of two work is the given a new constructive method for solving the problem of the behavior of geodesic on hyperbolic surfaces of genus g, k punctures and with n geodesic boundary components. At first:1) we obtain a complete classification of all possible geodesic curves on the simplest hyperbolic 2-manifolds (hyperbolic horn; hyperbolic cylinder; parabolic horn (cusp), hyperbolic pants); 2) on surface of genus 2; Finally: 3) on compact closed hyperbolic surface without boundarie (general case); 4) on hyperbolic surface of genus g and with n geodesic boundary components; 5) on hyperbolic 1-punctured torus; on generalized hyperbolic pants; in general case: for any punctured hyperbolic surface of genus g and k punctures; 6) in the most general case: or in any hyperbolic surface of genus g, k punctures and with n boundary curves.