In this paper we study the multifractal structure of Schramm's SLE curves. We derive the values of the (average) spectrum of harmonic measure and prove Duplantier's prediction for the multifractal spectrum of SLE curves. The spectrum can also be used to derive estimates of the dimension, Hölder exponent and other geometrical quantities. The SLE curves provide perhaps the only example of sets where the spectrum is non-trivial yet exactly computable.