The approach to equilibrium for systems of reaction-diffusion equations on bounded domains is studied geometrically. It is shown that equilibrium is approached via low-dimensional manifolds in the infinite-dimensional function space for these dissipative, parabolic systems. The fundamental aspects of this process are mapped Out in some detail for single species cases and for two-species cases where there is an exact solution. It is shown how the manifolds reduce the dimensionality of the system from infinite dimensions to only a few dimensions.