(Universal) enveloping algebras of finite-dimensional Lie algebras are some of the most well-understood examples in noncommutative ring theory. On the other hand, enveloping algebras of infinite-dimensional Lie algebras are much more mysterious . This talk will survey what's known about these rings, focussing on noetherianity questions. In particular, we'll describe the use of noncommutative algebraic geometry techniques to prove that that the enveloping algebra of the Witt algebra of vector fields on the afifine line is not noetherian.