This one of the courses at the Lipa Summer School.
A recognisable language of words is one that can be recognised by a monoid or a finite automaton. Similar statements hold for trees, and various infinite extensions of words and trees. What about graphs? I will discuss this in my talk, which is based mainly on ideas of Courcelle. There are at least two algebras for graphs (corresponding to graph parameters like treewidth or cliquewidth). I will also discuss the connections to monadic second-order logic, i.e. Courcelle’s conjecture.