Hopf algebras provide a rich framework for studying symmetry in noncommutative settings, generalizing both group algebras and enveloping algebras of Lie algebras. If one has any hope of understanding the structure of noetherian Hopf algebras, one should first understand when group algebras and enveloping algebras are noetherian. These two famously difficult problems are open to this day. In this talk, I will give a survey of what is known about the noetherianity of group algebras and enveloping algebras, mainly focusing on the Lie algebra side.