  Non-linear sigma models with extended supersymmetry have constrained target
space geometries, and can serve as effective tools for investigating and
constructing new geometries. Analyzing the geometrical and topological
properties of sigma models is necessary to understand the underlying structures
of string theory. The most general two-dimensional sigma model with manifest
N=(2,2) supersymmetry can be parametrized by chiral, twisted chiral and
semichiral superfields. In the research presented in this thesis, N=(4,4)
(twisted) supersymmetry is constructed for a semichiral sigma model. It is
found that the model can only have additional supersymmetry off-shell if the
target space has a dimension larger than four. For four-dimensional target
manifolds, supersymmetry can be introduced on-shell, leading to a hyperk\"ahler
manifold, or pseudo-supersymmetry can be imposed off-shell, implying a target
space which is neutral hyperk\"ahler.
  Different sigma models and corresponding geometries can be related to each
other by T-duality, obtained by gauging isometries of the Lagrangian. The
semichiral vector multiplet and the large vector multiplet are needed for
gauging isometries mixing semichiral superfields, and chiral and twisted chiral
superfields, respectively. We find transformations that close off-shell to a
N=(4,4) supersymmetry on the field strengths and gauge potentials of the
semichiral vector multiplet, and show that this is not possible for the large
vector multiplet.
  A sigma model parametrized by chiral and twisted chiral superfields can be
related to a semichiral sigma model by T-duality. The N=(4,4) supersymmetry
transformations of the former model are linear and close off-shell, whereas
those of the latter are non-linear and close only on-shell. We show that this
discrepancy can be understood from T-duality, and find the origin of the
non-linear terms in the transformations.
