  Entropy and the fluctuation-dissipation theorem are at the heart of
statistical mechanics near equilibrium. Driving a system beyond the linear
response regime leads to (i) the breakdown of the fluctuation-dissipation
theorem and (ii) a nonzero entropy production rate. We show how both phenomena
are related using the general framework of stochastic thermodynamics suitable
for soft matter systems governed by stochastic dynamics and driven through
nonconservative forces or external flows. In particular, the excess of the
fluctuation-dissipation theorem in a nonequilibrium steady state compared to
equilibrium is related to total entropy production. Alternative recent
derivations of generalized fluctuation-dissipation theorems are sketched and
related to each other. The theory is illustrated for two systems: a driven
single colloidal particle and systems driven through simple shear flow.
