We study different physical, chemical, or biological processes involving replication, transformation, and disappearance processes, as well as transport processes, and assume that the time and space dependence of the species densities are known. We derive two types of Fisher equations. The first type relates the average value of the time derivative of the relative time-specific rates of growth of the different species to the variance of the relative, time-specific rates of growth. A second type relates the average value of the gradient or the divergence of the relative, space-specific rates of growth to the space correlation matrix of the relative, space-specific rates of growth. These Fisher equations are exact results, which are independent of the detailed kinetics of the process: they are valid whether the evolution equations are linear or nonlinear, local or nonlocal in space and͞or time and can be applied for the study of a large class of physical, chemical, and biological systems described in terms of time-and͞or spacedependent density fields. We examine the implications of our generalized Fisher relations in population genetics, biochemistry, and chemical kinetics (reaction-diffusion systems). We show that there is a connection between the enhanced (hydrodynamic) transport of mutations induced by population growth and space-specific rate vectors: the velocity of enhanced transport is proportional to the product of the diffusion coefficient of the species and the space rate vector; this relation is similar to a fluctuation-dissipation relation in statistical mechanics. F isher's fundamental theorem of natural selection is one of the basic laws of population genetics. In 1930, Fisher showed that for single-locus genetic systems with pure selection and constant selection coefficients, the rate of variation of the average population fitness equals the genetic variance of the fitness (1). Because the variance is nonnegative, it follows that for systems with pure selection and constant rate coefficients, the average fitness always increases in time, a result that is compatible with general ideas of biological evolution. Fisher claimed that this law should hold the same position among biological sciences as the second law of thermodynamics in physical sciences (1). Many generalizations of Fisher's theorem have been suggested in the literature. A first generalization was introduced by Fisher himself, by taking into account the effects of overpopulation and the effect of deterioration of the environment (1). Further generalizations were introduced by Wright (2), Kimura (3), Price (4), Frank (5) and Ewens (6), Lessard (7), Crow (8), Nagylaki (9), and Edwards (10). A recent article in PNAS (11) reports on a general f luctuationdissipation relation for biological systems, which is qualitatively similar to the Fisher equation. Fisher's theorem also was used for the analysis of economic competition (12).Our research on Fisher's theorem was stimulated by a number of different problems in chemistry and biology. We noticed...