  Diffusion of small particles is omnipresent in a plentiful number of
processes occurring in Nature. As such, it is widely studied and exerted in
almost all branches of sciences. It constitutes such a broad and often rather
complex subject of exploration that we opt here to narrow down our survey for
the case of the diffusion coefficient for a Brownian particle which can be
modeled in the framework of Langevin dynamics. Our main focus will center on
the temperature dependence of the diffusion coefficient for several fundamental
models of diverse physical systems. Starting out with diffusion in equilibrium
for which the Einstein theory holds we consider a number of physical situations
away from free Brownian motion and end with surveying nonequilibrium diffusion
for a time-periodically driven Brownian particle dwelling randomly in a
periodic potential. For this latter situation the diffusion coefficient
exhibits an intriguingly non-monotonic dependence on temperature.
