This paper is devoted to the classification of all geodesic curves on 2-dimensional hyperbolic manifolds (the global behavior) by using a new constructive method - generalized colored multilateral method. We start by considering this problem on hyperbolic pants in hopes of discovering a method which can be easily generalized to the problem of behavior of geodesic on any hyperbolic surface. 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. To facilitate the understanding and further description, we agree to call the sides of the six-rectangle (hyperbolic right-angled hexagon) black, if they are obtained from boundary geodesic circles of pants, and the other three sides we agree to consider painted in different colors (for example, red, blue and green straight). The study 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 generalized pants, on right hexagons, etc.) and the behavior of geodesics for each piece is solved on it, and then the result of the investigation returns (by gluing) onto the original surface. To summarize what has been said, we can conclude that a concrete method of investigating the behavior of geodesics on hyperbolic 2-manifolds is based on the idea of preliminary research on these pieces (on the set of hyperbolic pants and their generalized), in the subsequent consolidation of research results using the method of generalized colored multilateral. In more details, the following main results of the study were obtained. On hyperbolic pants, we classified all possible type of behavior of geodesic lines, based on the algorithm for constructing the corresponding system of colored angles, and by the sides parallel to the considered side of the generalized multilateral obtained from a hyperbolic hexagon. Further, the concept of the category of angles is introduced, and with the help of these categories an algorithm for recognizing the type of a geodesic is given. The main purpose of this paper is to give a new constructive method for solving the problem of the behavior of geodesic on an arbitrary hyperbolic surface of signature (g; n; k). Such a compressed formulated result can be disclosed as follows. For this purpose, with the help of proposed practical approach 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)); 2) we describe the behavior of geodesics in the following cases: on a surface of genus 2; on an compact closed hyperbolic surface without boundary (general case); on hyperbolic surface of genus g and with n geodesic boundary components; on hyperbolic 1-punctured torus; on generalized hyperbolic pants; in general case: for any (oriented) punctured hyperbolic surface M of genus g and k punctures; in the most general case: for any hyperbolic surface of signature (g; n; k) (with genus g, n boundary components and k cusps).