  Classical gradient systems have a linear relation between rates and driving
forces. In generalized gradient systems we allow for arbitrary relations
derived from general non-quadratic dissipation potentials. This paper describes
two natural origins for these structures.
  A first microscopic origin of generalized gradient structures is given by the
theory of large-deviation principles. While Markovian diffusion processes lead
to classical gradient structures, Poissonian jump processes give rise to
cosh-type dissipation potentials.
  A second origin arises via a new form of convergence, that we call
EDP-convergence. Even when starting with classical gradient systems, where the
dissipation potential is a quadratic functional of the rate, we may obtain a
generalized gradient system in the evolutionary $\Gamma$-limit. As examples we
treat (i) the limit of a diffusion equation having a thin layer of low
diffusivity, which leads to a membrane model, and (ii) the limit of diffusion
over a high barrier, which gives a reaction-diffusion system.
