  In 1931/32, Schroedinger studied a hot gas Gedankenexperiment, an instance of
large deviations of the empirical distribution and an early example of the
so-called maximum entropy inference method. This so-called Schroedinger bridge
problem (SBP) was recently recognized as a regularization of the
Monge-Kantorovich Optimal Mass Transport (OMT), leading to effective
computation of the latter. Specifically, OMT with quadratic cost may be viewed
as a zero-temperature limit of SBP, which amounts to minimization of the
Helmholtz's free energy over probability distributions constrained to possess
given marginals. The problem features a delicate compromise, mediated by a
temperature parameter, between minimizing the internal energy and maximizing
the entropy. These concepts are central to a rapidly expanding area of modern
science dealing with the so-called {\em Sinkhorn algorithm} which appears as a
special case of an algorithm first studied by the French analyst Robert Fortet
in 1938/40 specifically for Schroedinger bridges. Due to the constraint on
end-point distributions, dynamic programming is not a suitable tool to attack
these problems. Instead, Fortet's iterative algorithm and its discrete
counterpart, the Sinkhorn iteration, permit computation by iteratively solving
the so-called {\em Schroedinger system}. In both the continuous as well as the
discrete-time and space settings, {\em stochastic control} provides a
reformulation and dynamic versions of these problems. The formalism behind
these control problems have attracted attention as they lead to a variety of
new applications in spacecraft guidance, control of robot or biological swarms,
sensing, active cooling, network routing as well as in computer and data
science. This multifacet and versatile framework, intertwining SBP and OMT,
provides the substrate for a historical and technical overview of the field
taken up in this paper.
