In this thesis we prove an equivariant version of the Jeffrey-Kirwan localization theorem for non-compact symplectic and hyper-Kähler quotients. In the non-compact setting the integrals are defined by the Atiyah-Bott-Berline-Vergne formula. We introduce an equivariant version of the Jeffrey-Kirwan residue. As applications, we compute the cohomology ring of the Hilbert scheme of points on the plane in a new way, moreover we also compute Nekrasov's partition function on the framed moduli space of torsion free sheaves on the complex projective plane.