  Lagrangian reduction by stages is used to derive the Euler-Poincar\'e
equations for the nondissipative coupled motion and micromotion of complex
fluids. We mainly treat perfect complex fluids (PCFs) whose order parameters
are continuous material variables. These order parameters may be regarded
geometrically either as objects in a vector space, or as coset spaces of Lie
symmetry groups with respect to subgroups that leave these objects invariant.
Examples include liquid crystals, superfluids, Yang-Mills magnetofluids and
spin-glasses. A Lie-Poisson Hamiltonian formulation of the dynamics for perfect
complex fluids is obtained by Legendre transforming the Euler-Poincare
formulation. These dynamics are also derived by using the Clebsch approach. In
the Hamiltonian and Lagrangian formulations of perfect complex fluid dynamics
Lie algebras containing two-cocycles arise as a characteristic feature.
  After discussing these geometrical formulations of the dynamics of perfect
complex fluids, we give an example of how to introduce defects into the order
parameter as imperfections (e.g., vortices) that carry their own momentum. The
defects may move relative to the Lagrangian fluid material and thereby produce
additional reactive forces and stresses.
