The space of marked groups is a compact Polish space that parameterizes all countable groups. This space allows for tools from descriptive set theory to be applied to study group-theoretic questions. The class of elementary amenable groups is the smallest class that contains the abelian groups and the finite groups and that is closed under group extension, taking subgroups, taking quotients, and taking countable directed unions. In this talk, we first give a characterization of elementary amenable marked groups in terms of well-founded trees. We then show the set of elementary amenable marked groups is coanalytic and non-Borel. This gives a new proof of a theorem of Grigorchuk: There are finitely generated amenable non-elementary amenable groups.