  We present a systematic study of ${\cal N}=(2,2)$ supersymmetric non-linear
sigma models on $S^2$ with the target being a K\"ahler manifold. We discuss
their reformulation in terms of cohomological field theory. In the
cohomological formulation we use a novel version of 2D self-duality which
involves a $U(1)$ action on $S^2$. In addition to the generic model we discuss
the theory with target space equivariance corresponding to a supersymmetric
sigma model coupled to a non-dynamical supersymmetric background gauge
multiplet. We discuss the localization locus and perform a one-loop calculation
around the constant maps. We argue that the theory can be reduced to some
exotic model over the moduli space of holomorphic disks.
