We give a summary of some of the existing work on coalgebra and coinduction: a general introduction to the coalgebraic framework is given, focussing especially on the instance of coalgebraic representations of regular and context-free languages, and deterministic automata. We show how Brzozowski derivatives of regular expressions fit into the coalgebraic framework, and present Rutten's coalgebraic version of Kleene's theorem. Finally, we present some recent results on our attempts to extend this coalgebraic approach towards context-free languages and grammars.