  We give a review of the systematic construction of hierarchies of soliton
flows and integrable elliptic equations associated to a complex semi-simple Lie
algebra and finite order automorphisms. For example, the non-linear
Schr\"odinger equation, the n-wave equation, and the sigma-model are soliton
flows; and the equation for harmonic maps from the plane to a compact Lie
group, for primitive maps from the plane to a $k$-symmetric space, and constant
mean curvature surfaces and isothermic surfaces in space forms are integrable
elliptic systems. We also give a survey of
  (i) construction of solutions using loop group factorizations,
  (ii) PDEs in differential geometry that are soliton equations or elliptic
integrable systems,
  (iii) similarities and differences of soliton equations and integrable
elliptic systems.
