  We calculate the vector and scalar form factors of the pion to two loops in
Chiral Perturbation Theory. We estimate the unknown O(p^6) constants using
resonance exchange. We make a careful comparison to the available data and
determine two O(p^4) constants rather precisely, and two O(p^6) constants less
precisely. We also use Chiral Perturbation Theory to two loops to extract in a
model--independent manner the charge radius of the pion from the available
data, and obtain \rpiV=0.437\pm 0.016 fm^2.
