  We present and apply a theory of one parameter $C_0$-semigroups of linear
operators in locally convex spaces. Replacing the notion of equicontinuity
considered by the literature with the weaker notion of sequential
equicontinuity, we prove the basic results of the classical theory of
$C_0$-equicontinuous semigroups: we show that the semigroup is uniquely
identified by its generator and we provide a generation theorem in the spirit
of the celebrated Hille-Yosida theorem. Then, we particularize the theory in
some functional spaces and identify two locally convex topologies that allow to
gather under a unified framework various notions $C_0$-semigroup introduced by
some authors to deal with Markov transition semigroup. Finally, we apply the
results to transition semigroups.
