  We consider the evolutionary trajectories traced out by an infinite
population undergoing mutation-selection dynamics in static, uncorrelated
random fitness landscapes. Starting from the population that consists of a
single genotype, the most populated genotype \textit{jumps} from a local
fitness maximum to another and eventually reaches the global maximum. We use a
strong selection limit, which reduces the dynamics beyond the first time step
to the competition between independent mutant subpopulations, to study the
dynamics of this model and of a simpler one-dimensional model which ignores the
geometry of the sequence space. We find that the fit genotypes that appear
along a trajectory are a subset of suitably defined fitness \textit{records},
and exploit several results from the record theory for non-identically
distributed random variables. The genotypes that contribute to the trajectory
are those records that are not \textit{bypassed} by superior records arising
further away from the initial population. Several conjectures concerning the
statistics of bypassing are extracted from numerical simulations. In
particular, for the one-dimensional model, we propose a simple relation between
the bypassing probability and the dynamic exponent which describes the scaling
of the typical evolution time with genome size. The latter can be determined
exactly in terms of the extremal properties of the fitness distribution.
