Hilbert's 5th problem, in its most basic form, asks if every compact topological group, which admits the structure of a smooth manifold, is a Lie group. In this form, it was answered affirmatively by von Neumann in 1929. If one takes a homotopical interpretation of the word "admits", the question is more subtle, and one is led to the notion of a finite loop space. These turn out not quite to be Lie groups, but nevertheless posses a rich enough structure to admit a classification. My talk will outline this story, which starts with a 1941 paper of Hopf: "Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen" and ends close to the present.