  This paper focuses on the maximum speed at which biological evolution can
occur. I derive inequalities that limit the rate of evolutionary processes
driven by natural selection, mutations, or genetic drift. These \emph{rate
limits} link the variability in a population to evolutionary rates. In
particular, high variances in the fitness of a population and of a quantitative
trait allow for fast changes in the trait's average. In contrast, low
variability makes a trait less susceptible to random changes due to genetic
drift. The results in this article generalize Fisher's fundamental theorem of
natural selection to dynamics that allow for mutations and genetic drift, via
trade-off relations that constrain the evolutionary rates of arbitrary traits.
The rate limits can be used to probe questions in various evolutionary biology
and ecology settings. They apply, for instance, to trait dynamics within or
across species or to the evolution of bacteria strains. They apply to any
quantitative trait, e.g., from species' weights to the lengths of DNA strands.
