  This thesis considers one and two dimensional supersymmetric nonlinear sigma
models. First there is a discussion of the geometries of one and two
dimensional sigma models, with rigid supersymmetry.
  For the one-dimensional case, the supersymmetry is promoted to a local one
and the required gauge fields are introduced. The most general Lagrangian,
including these gauge fields, is found. The constraints of the system are
analysed, and its Dirac quantisation is investigated.
  In the next chapter we introduce equivariant cohomology which is used later
in the thesis.
  Then actions are constructed for (p,0)- and (p,1)- supersymmetric, $1 \leq p
\leq 4$, two-dimensional gauge theories coupled to non-linear sigma model
matter with a Wess-Zumino term.
  The scalar potential for a large class of these models is derived. It is then
shown that the Euclidean actions of the (2,0) and (4,0)-supersymmetric models
without Wess-Zumino terms are bounded by topological charges which involve the
equivariant extensions of the Kahler forms of the sigma model target spaces
evaluated on the two-dimensional spacetime.
  Similar bounds for Euclidean actions of appropriate gauge theories coupled to
non-linear sigma model matter in higher spacetime dimensions are given which
now involve the equivariant extensions of the Kahler forms of the sigma model
target spaces and the second Chern character of gauge fields. It is found that
the BPS configurations are generalisations of abelian and non-abelian vortices.
