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Field theoretical renormalization group methods are applied to the Obukhov-Kraichnan model of a passive scalar advected by the Gaussian velocity field with the covariance ͗v(t,x)v(tЈ,x)͘Ϫ͗v(t,x)v(tЈ,xЈ)͘ϰ␦(t ϪtЈ)͉xϪxЈ͉ . 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 certain essential or ''dangerous'' composite operators ͓powers of the local dissipation rate͔, whose negative critical dimensions determine anomalous exponents. The main technical result is the calculation of the anomalous exponents in the order 2 of the expansion. Generalization of the results obtained to the case of a ''slow'' velocity field is also presented.

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The field theoretic renormalization group is applied to the stochastic Navier-Stokes equation that describes fully developed fluid turbulence in d > 2 dimensions. For the first time, the complete two-loop calculation of the renormalization constant, the β function, the fixed point and the ultraviolet correction exponent is performed. The Kolmogorov constant and the inertial-range skewness factor are expressed in terms of universal (in the sense of the theory of critical behavior) quantities, which allows one to construct for them a regular perturbative calculational scheme. The practical calculations are performed up to the second order of the ε-expansion (two-loop approximation). The results obtained are in a good agreement with the experiment. The large d behavior of these quantities is briefly discussed. The possibility of the extrapolation of the ε-expansion beyond the threshold where the sweeping effects become important is demonstrated by the example of a Galilean-invariant quantity, the equal-time pair correlation function of the velocity field.