The structure of quantum corrections and exact results in supersymmetric theories revealed by the higher covariant derivative regularizaton

Some recent results of investigating quantum corrections in supersymmetric theories revealed with the help of the Slavnov's higher covariant derivative regularization are reviewed. In particular, we demonstrate that the betafunction of N=1 supersymmetric theoires is related to the anomalous dimensions of the matter superfields by the NSVZ relation if a theory is regularized by higher covariant derivatives and the renormalization group functions are defined in terms of the bare couplings, because the corresponding loop corrections are given by integrals of double total derivatives in the monentum space. For the standard renormalization group functions we show that an all loop NSVZ renormalization scheme is given by the HD+MSL renormalization prescription, when the higher covariant derivative regularization is supplemented by minimal subtraction of logarithms. Applications of these results to the precise calculations in various supersymmetric theories are briefly described.