  This paper concerns constructing topological sigma models governing maps from
semirigid super Riemann surfaces to general target supermanifolds. We define
both the A model and B model in this general setup by defining suitable BRST
operators and physical observables. Using supersymmetric localization, we
express correlation functions in these theories as integrals over suitable
supermanifolds. In the case of the A model, we obtain an integral over the
supermoduli space of "superinstantons". The language of supergeometry is used
extensively throughout this paper.
