Abstract: A "frieze" is an infinite strip of numbers satisfying certain determinantal identities, or any one of several generalizations of this idea. In this talk, I will give an introduction to friezes whose shape is determined by a Dynkin diagram (motivated by their exceptional properties as well as connections to representation theory and cluster algebras). One of the simplest questions was open until recently: are there finitely many such friezes? Time permitting, I will describe two proofs of finiteness, and on-going work into enumerating them.