We bound the Pythagoras number of a real projective subvariety: the smallest positive integer r such that every sum of squares of linear forms in its homogeneous coordinate ring is a sum of a most r squares. We will describe three distinct upper bounds involving known invariants. In contrast, our lower bound depends on a new invariant called quadratic persistence. This talk is based on joint work with Greg Blekherman, Rainer Sinn, and Mauricio Velasco.