  This paper develops the theory of affine Euler-Poincar\'e and affine
Lie-Poisson reductions and applies these processes to various examples of
complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids
and superfluids, spin glasses, microfluids, and liquid crystals. As a
consequence of the Lagrangian approach, the variational formulation of the
equations is determined. On the Hamiltonian side, the associated Poisson
brackets are obtained by reduction of a canonical cotangent bundle. A
Kelvin-Noether circulation theorem is presented and is applied to these
examples.
