The field theoretic renormalization group is applied to Kraichnan's model of a passive scalar quantity advected by the Gaussian velocity field with the pair correlation function ∝ δ(t - t′)/κ^d+ε. Inertial-range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of "dangerous" composite operators (powers of the local dissipation rate), whose negative critical dimensions determine anomalous exponents. The latter are calculated to order ε³ of the ε expansion (three-loop approximation).