  A variety of astronomical phenomena appear to not satisfy the ergodic
hypothesis in the relevant stationary state, if any. As such, there is no
reason for expecting the applicability of Boltzmann-Gibbs (BG) statistical
mechanics. Some of these phenomena appear to follow, instead, nonextensive
statistical mechanics. In the same manner that the BG formalism is based on the
entropy $S_{BG}=-k \sum_i p_i \ln p_i$, the nonextensive one is based on the
form $S_q=k(1-\sum_ip_i^q)/(q-1)$ (with $S_1=S_{BG}$). The stationary states of
the former are characterized by an {\it exponential} dependence on the energy,
whereas those of the latter are characterized by an (asymptotic) {\it
power-law}. A brief review of this theory is given here, as well as of some of
its applications, such as the solar neutrino problem, polytropic
self-gravitating systems, galactic peculiar velocities, cosmic rays and some
cosmological aspects. In addition to these, an analogy with the Keplerian
elliptic orbits {\it versus} the Ptolemaic epicycles is developed, where we
show that optimizing $S_q$ with a few constraints is equivalent to optimizing
$S_{BG}$ with an infinite number of constraints.
