Universal enveloping algebras of finite-dimensional Lie algebras are fundamental examples of well-behaved noncommutative rings, yet enveloping algebras of infinite-dimensional Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but we are very far from being able to prove this in complete generality. In this talk, I will focus on my recent work with Jason Bell on the noetherianity of enveloping algebras of Lie algebras of derivations. I will mainly talk about our classification of subalgebras of the (one-sided) Witt algebra, which implies the non-noetherianity of their enveloping algebras as a direct corollary.