Let G be a simple Lie group over C and let Lie(G) be its Lie algebra. The exterior algebra of Lie(G) is extensively studied in literature for its link with the geometry of G and the combinatorics of the Weyl group W(G). In this talk I will present an overview of some classical results about the exterior algebra in a "representation theory"-flavour, with a particular attention to two open conjectures due to Kostant and Reeder.