What's so great about the Archimedean screw? Well, for one thing, it's affine homogeneous as a surface in $R^3$. The Cayley surface is another classical example. Using a Lie algebraic approach, the affine homogeneous surfaces in $R^3$ were classified in 1996 by Doubrov, Komrakov, and Rabinovich. I shall describe a geometric approach of Vladimir Ezhov and myself, which provides an alternative classification in $R^3$ and some further classifications in $R^4$ and $C^4$.