The paper studies maximal surfaces (with degenerate points) in the 3-dimensional Minkowski space, i.e. space-like surfaces with zero mean curvature. The author analyzes the Weierstrass-type representation corresponding to a maximal map with a given simple end. The main result of the paper gives a characterization of the asymptotic ends of such surfaces. Namely, besides those ends that are asymptotic to a space-like plane or a null line, there are three other possible types, and parametrizations are explicitly given.