  We introduce the concept evolutionary semigroups on path spaces, generalizing
the notion of transition semigroups to possibly non-Markovian stochastic
processes. We study the basic properties of evolutionary semigroups and, in
particular, prove that they always arise as the composition of the shift
semigroup and a single operator called the expectation operator of the
semigroup. We also prove that the transition semigroup of a Markov process can
always be extended to an evolutionary semigroup on the path space whenever the
Markov process can be realized with the appropriate path regularity. As first
examples of evolutionary semigroups associated to non-Markovian processes, we
discuss deterministic evolution equations and stochastic flows driven by L\'evy
processes. The latter in particular include certain stochastic delay equations.
