We are concerned in this paper with the behavior in the large of the geodesic lines on hyperbolic manifolds, giving special atention to the two-dimensional cases. A geodesic in a hyperbolic surfaces is an arc which in the local coordinate charts, is the image of a geodesic arc of the hyperbolic plane. Topological surfaces are often thought of as the result of pasting togetheter polygons. Provided you have enough topology, pants decompositions are a natural way of decomposing (orientable) surfaces or conversely one can build a surface by pasting 3 holed spheres (pants) a long their cufs. So it is important to study the behavior of geodesics on a pair of pants. Thanks to the development of the new constructive approach, in this paper, the author succeeded to receive “in a certain sense” the solution for the behavior of the geodesics in general on the hyperbolic manifolds, structure of geodesics and their types. Arbitrary hyperbolic surfaces M, closed or open, of finite or infinite genus are considered. Yet another way to define a hyperbolic surface is via its universal cover. For the behavior of the geodesics on the specified fragments (hyperbolic pants, etc.) it is used a certain figure, named in the text of the work the multilateral. The study of the behavior of the geodesics in this paper is being carried out gradually, in order of collecting the surface, the reverse order of cutting the surface into fragments (i.e. pants). The surface is cut into typical pieces (for example, on pants or their degenerations, on right hexagons, etc.) and the question of the behavior of the geodesics for each piece is solved on it, and then the result of the investigation returns (by gluing) onto the original surface. With the help of these multilaterals, it is possible to determine the nature of the behavior of the geodesics on the surface. Any given hyperbolic (closed, i.e., ordinary) surface can be cut into pants and the question is how, when gluing such pants, connect them on a common surface. But it may seem (when gluing of the surface from the pants is not finished yet) that the surface of genus g has also n components (the surface has a geodesic boundary). And, going further, we notice that the boundary of the surface can degenerate: transform into cuspidal ends (cusps) and into conical points. Thus, we arrive at the most general case, the surfaces of the signature (g,n,k), the preliminary investigation of the behavior of the geodesics on these pieces. A concrete method of investigating the behavior of the geodesics on hyperbolic 2-manifolds is based on the idea of preliminary research on these pieces (on the set of hyperbolic pants and their degenerations), in the subsequent consolidation of research results using the method proposed in this paper (sometimes called the method of generalized coloured multilaterals). The solution is based on the study of the behavior of the geodesics on the simplest hyperbolic surfaces (hyperbolic pants, degenerate hyperbolic pants, thrice-punctured sphere, etc.), some of which have long attracted the attention of geometers.