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∑_ip_i ln p_i, the nonextensive one is based on the form S_q = k(1 −∑_ip_i^q)/(q− 1) (with S₁ = S_BG). The stationary states of the former are characterized by an exponential dependence on the energy, whereas those of the latter are characterized by an (asymptotic) 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 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.Facultad de Ciencias Astronómicas y Geofísicas