SOL is the 3–dimensional solvable Lie group, the strangest of the 8 Thurston geometries. I will sketch a proof that the metric spheres in SOL are all topological spheres, smooth for radius < π √2 and for larger radii smooth away from 4 singular planar arcs. This result is in turn a consequence of a precise characterization of the cut-locus of SOL. Part of my talk will be a computer demo showing the results and part will be an exposition of some of the beautiful discoveries of Matt Grayson about the geodesics in SOL, which are used in the proof.