Let F be a p-adic field and G the F-points of a reductive group over F. If U is a compact open subgroup of G, and R is a ring, then the Hecke algebra attached to U, G, and R is the space of compactly supported R-valued functions on G that are left and right U-invariant; convolution gives this space the structure of an R-algebra. When R is a field of characteristic zero, a celebrated result of Bernstein shows that such Hecke algebras have good finiteness properties; in particular they are Noetherian. For R a more complicated ring such as Z_{\ell}, the question has been open for over thirty years. I will explain how recent results of Fargues and Scholze can be used to resolve this question.