We give a brief introduction to the coalgebraic method and show two recent developments that attest its generality and usefulness.

We first show a new proof of Brzozowski's minimization algorithm and discuss generalizations resulting from this new perspective.

We then present CoCaml, a functional programming language extending OCaml, which allows us to define functions on coinductive datatypes parameterized by an equation solver. We provide compelling examples that attest to the usefulness of the new programming constructs.