Asymptotic Behavior of Covering Curves on the Universal Coverings of Surfaces

To date, a large number of publications have appeared that are devoted to the study of asymptotic properties of the lifts of curves without self-intersections to the universal covering and the “collation” of these curves (in a certain sense) with lines of constant geodesic curvature that have the same asymptotic direction as the curves under investigation. This paper contains a survey of the results obtained. The ideas of proofs for the main results and the sketches of constructions for important examples on this subject field are presented.