We review the variational principles called Maximum Entropy (Max Ent) and Maximum Caliber (Max Cal). Max Ent originated in the statistical physics of Boltzmann and Gibbs, as
a theoretical tool for predicting the equilibrium states of thermal systems. Later, entropy maximization was also applied to matters of information, signal transmission and
image reconstruction. Recently, since the work of Shore and Johnson, Max Ent has been regarded as a principle that is broader than either just physics or information alone.
Max Ent is a procedure that ensures that inferences drawn from stochastic data satisfy basic self-consistency requirements. We review the different historical justifications
for the entropy, S = -i pi logpi, and its corresponding variational principles. As an illustration of the broadening purview of maximum entropy principles, we also review
Maximum Caliber, which is path entropy maximization applied to the trajectories of dynamical systems. We give examples in which Maximum Caliber is used to interpret dynamical
fluctuations in biology and on the nanoscale, in single-molecule and few-particle systems such as molecular motors, chemical reactions, biological feedback circuits
and diffusion in microfluidics devices.