I will give a survey on my recent work with Bruno Courcelle concerning the expressive power of monadic second-order logic on certain graph classes. In particular, I will present our work on the transduction hierarchy, a classification of graph classes according to their encoding power with respect to transductions. I will also give an overview on more recent work on the definability of linear orderings on certain classes of graphs. Here we aim at simple combinatorial characterisations of whether such an ordering is definable or not.