Conjecturally, under suitable assumptions there exists a morphism of cohomological field theories from the equivariant Gromov-Witten theory of M, w to the Gromov-Witten theory of the symplectic quotient. The idea, due to D.A. Salamon, is to define such a deformation by counting gauge equivalence classes of symplectic vortices over the complex plane C. The present memoir is part of a project whose goal is to make this definition rigorous. Its main results deal with the symplectically aspherical case.