universal enveloping algebras of finite-dimensional lie algebras are fundamental examples of well-behaved noncommutative rings. on the other hand, enveloping algebras of infinite-dimensional lie algebras remain mysterious. for example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. in this talk, i will summarize what is known about the noetherianity of enveloping algebras, with a focus on lie algebras of derivations of associative algebras.