The study of Markov processes on complete separable metric spaces \((E,d)\) naturally leads to transition semigroups on the space \(C_b(E)\) which are not strongly continuous with respect to the norm of this space. The authors consider locally equi-continuous strongly continuous semigroups on locally convex spaces \((X,\tau)\) which are also equipped with a suitable auxiliary norm. Let \(\mathcal{N}\) denote the set of \(\tau\)-continuous semi-norms which are bounded by the norm. If \((X,\tau)\) has the property that \(\mathcal{N}\) is closed under countable convex combinations, then a number of results carry over from the Banach space setting to this new setting in a straightforward way. In particular, the authors extend the important Hille-Yosida theorem. The authors go on to relate their results to those which hold for bi-continuous semigroups, showing that, if \(E\) is a Polish space and \(H\) is a Hilbert space, then these results can be applied to semigroups on \((C_b(E),\beta)\) and \((\mathcal{B}(H),\beta)\), where \(\beta\) denotes the respective strict topologies.