Abstract: First, I give a basic introduction to finite noncommutative spaces, and then I focus on the classification of finite real spectral triples.

Abstract: Hilbert's Fourteenth Problem asks about the finite generation of certain commutative rings. Furthermore, Hilbert's question was a major catalyst in the development of geometric invariant theory. But the basic question here makes sense more generally. I will discuss examples, successes, myths, and new approaches.