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A variety of astronomical phenomena appear to not satisfy the ergodichypothesis in the relevant stationary state, if any. As such, there is noreason for expecting the applicability of Boltzmann-Gibbs (BG) statisticalmechanics. Some of these phenomena appear to follow, instead, nonextensivestatistical mechanics. In the same manner that the BG formalism is based on theentropy $S_{BG}=-k \sum_i p_i \ln p_i$, the nonextensive one is based on theform $S_q=k(1-\sum_ip_i^q)/(q-1)$ (with $S_1=S_{BG}$). The stationary states ofthe former are characterized by an {\it exponential} dependence on the energy,whereas those of the latter are characterized by an (asymptotic) {\itpower-law}. A brief review of this theory is given here, as well as of some ofits applications, such as the solar neutrino problem, polytropicself-gravitating systems, galactic peculiar velocities, cosmic rays and somecosmological aspects. In addition to these, an analogy with the Keplerianelliptic orbits {\it versus} the Ptolemaic epicycles is developed, where weshow that optimizing $S_q$ with a few constraints is equivalent to optimizing$S_{BG}$ with an infinite number of constraints.Comment: 23 pages including one EPS figure. To appear in the Proceedings of  the XIth United Nations / European Space Agency Workshop on Basic Space  Sciences, Office for Outer Space Affairs / United Nations (Cordoba, 9-13  September 2002), ed. H. Haubold, special issue of Astrophysics and Space  Science (Kluwer Academic Publishers, Dordrecht, 2003). It includes a reply to  the criticism by M. Nauenberg, cond-mat/0210561 [version 2

Nonextensive Statistical Mechanics: Some Links with Astronomical Phenomena