I will describe a way to regularize all the binary collisions and blow-up the triple collision singularity in the planar three-body problem, inspired by Lemaitre's work in the 1950's and 1960's. In the end, the regularizing map is a beautiful branched covering map of the Riemann sphere. Trying to do something similar for the four-body problem leads to the realm of complex algebraic surfaces. (Joint work with R. Montgomery).