An approach is proposed which allows consideration from common positions the self-similar and self-affine properties of river drainage basins and which introduces respective characteristics of their fractal structure. Horton's (1945) model has been assumed as the basis of this study. It is shown that self-similarity of river drainage basins is characterized by the self-similarity dimension DS = ln RB/ln RL. In the case of self-affinity, two scaling indices DL = ln RB/ln RL and Dw = ln RB/ln (RA/RL) are required, the combinations of which determine the Hurst H = DL/DW index (characterizing the degree of drainage basins self-affinity) and the lacunary dimension DG = 2DLDW/(DL + DW) (characterizing the degree of lacunarity or noncompactness of drainage basins). On the basis of analysis of published data from rivers in the United States, Italy, Russia, and Romania, the conclusion is drawn that river drainage basins are generally noncompact (DG < 2) and self-affine (H < 1) fractal objects. Besides, for a quantitative description of river drainage basins, a new parameter is introduced, which characterizes the degree to which a large catchment is filled with smaller ones.