An important challenge in Teichmuller theory is to classify the special subvarieties of the moduli space of Riemann surfaces which are totally geodesic for the Teichmuller metric. These totally geodesic varieties are rare, and their origins remain a mystery. Until recently, the only known non-elementary examples were curves. I will discuss the first surprising example of a totally geodesic algebraic surface. This work is joint with Curt McMullen and Alex Wright.