In analyzing the literature on the fractal nature of river networks one can recognize several points that require clarification or additional investigations. For instance, many interpretations, with different conclusions, have been given to the important empirical results found in the previous literature on river geomorphology. Similarly, the argument concerning a possible ''more likely'' value of the fractal dimensions of river networks has not yet found a convincing answer. In an attempt to shed more light on the topic, particularly on the latter point, we have reexamined the basic aspects related to self-similarity in river networks using uniquely the fundamental concepts of fractal geometry. This approach leads to relations for the determination of the fractal dimensions which hold exactly for deterministic self-similar trees and are compatible with related findings reported in literature. Results obtained from application of these relations to several river networks in southern Italy indicate very low variability around the average values of about 1.7, 1.5, and 1.1 for the fractal dimensions of the whole river network, of its topological equivalent, and of its individual streams, respectively. This outcome, in addition to comparable results taken from the literature, provides new arguments to the hypothesis that natural networks tend to have the same fractal dimensions. ¿ American Geophysical Union 1996