Asymptotic stability of uniform, nonequilibrium stationary states in multicomponent systems communicating everywhere with uniform surroundings is analyzed. Conventional diffusion with first-order reactions is mathematically equivalent. The systems need not be close to equilibrium. Conditions are given under which stability is completely determined by eigenvalues of the matrices of reaction constants and diffusion coefficients separately; commuting reaction and diffusion matrices is a special case. Reaction and diffusion may interact to produce instability where neither does by itself. Stability behavior with more than three components is analytically intractable, but a complete catalog for two components and representative results for three are presented. How to design in principle a three-component system having specified stability behavior is shown. Implications for n-component systems are discussed, especially systems that can be approximated by a number of weakly interacting subsystems.