A new method for analyzing the self-similarity and self-affinity of single-thread channels is proposed. It permits the determination of the fractal scaling exponents, of the characteristic scales, and the evaluation of the degree of anisotropy for self-similar fractal lines. Based upon the application of this method to the Dniester and Pruth rivers we established the self-similarity of the river pattern on small scales and the self-affinity on large scales. For these rivers we obtained the fractal scaling exponents, the characteristic scales, and the anisotropy parameters. A computer model has been developed which simulates river patterns whose fractal properties are close to the properties of natural objects. A generalized model of fractal behavior of natural rivers is proposed. On the basis of self-affinity of natural and simulated rivers on large scales, a hypothesis has been formulated which explains the violation of the dimension principle in the well-known relation between the river length and the catchment area. ¿ American Geophysical Union 1993