Given a semisimple Lie algebra g, to each nilpotent orbit in g one attaches the finite W-algebra, which is a subquotient of the enveloping algebra of g, as well as the affine W-algebra, which is a subquotient of the enveloping algebra of the loop algebra g((t)). Since the introduction of W-algebras into mathematics from physics in the late 80s, much has been understood about their highest weight representations, thanks to the work of many mathematicians, though some fairly basic questions remain open.
After surveying aspects of this story for nonexperts, we will discuss some results in progress which should in particular finish the problem of computing the characters of simple highest weight modules for finite and affine W-algebras in characteristic zero. This is a report on work in progress with Arakawa and Faergeman.