We solve N = 2 supersymmetric Yang-Mills theories for arbitrary classical gauge group, i.e. SU (N ), SO(N ), Sp(N ). In particular, we derive the prepotential of the low-energy effective theory, and the corresponding Seiberg-Witten curves. We manage to do this without resolving singularities of the compactified instanton moduli spaces.

We apply equivariant integration technique, developed in the context of instanton counting, to two dimensional N = 2 supersymmetric Yang-Mills models. Twisted superpotential for U(N ) model is computed. Connections to the four dimensional case are discussed. Also we make some comments about the eight dimensional model which manifests similar features.

N = 2 supersymmetric Yang-Mills theories for all classical gauge groups, that is, for SU (N ), SO(N ), and Sp(N ) is considered. The equations which define the Seiberg-Witten curve are proposed. In some cases they are solved. It is shown that for (almost) all models allowed by the asymptotic freedom the 1-instanton corrections which follows from these equations agree with the direct computations and with known results.

We investigate the possibility to extract Seiberg-Witten curves from the formal series for the prepotential, which was obtained by the Nekrasov approach. A method for models whose SeibergWitten curves are not hyperelliptic is proposed. It is applied to the SU(N ) model with one symmetric or antisymmetric representations as well as for SU(N 1 ) × SU(N 2 ) model with (N 1 , N 2 ) or (N 1 , N 2 ) bifundamental matter. Solution are compared with known results. For the gauge group product we have checked the instanton corrections which follow from our curves against direct instanton counting computations up to two instantons.