Reduction theory for mechanical systems with symmetry has its roots in the
classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, P
oincare, and others. The modern vision of mechanics includes, besides the t
raditional mechanics of particles and rigid bodies, field theories such as
electromagnetism, fluid mechanics, plasma physics, solid mechanics as well
as quantum mechanics, and relativistic theories, including gravity. Symmetr
ies in these theories vary from obvious translational and rotational symmet
ries to less obvious particle relabeling symmetries in fluids and plasmas,
to subtle symmetries underlying integrable systems. Reduction theory concer
ns the removal of symmetries and their associated conservation laws. Variat
ional principles, along with symplectic and Poisson geometry, provide funda
mental tools for this endeavor. Reduction theory has been extremely useful
in a wide variety of areas, from a deeper understanding of many physical th
eories, including new variational and Poisson structures, to stability theo
ry, integrable systems, as well as geometric phases. This paper surveys pro
gress in selected topics in reduction theory, especially those of the last
few decades as well as presenting new results on non-Abelian Routh reductio
n. We develop the geometry of the associated Lagrange-Routh equations in so
me detail. The paper puts the new results in the general context of reducti
on theory and discusses some future directions. (C) 2000 American Institute
of Physics. [S0022-2488(00)02006-5].