  We review recent developments in the context of two-dimensional conformally
invariant sigma-models. These quantum field theories play a prominent role in
the covariant superstring quantization in flux backgrounds and in the analysis
of disordered systems.
  We present supergroup WZW models as primary examples of logarithmic conformal
field theories, whose structure is almost entirely determined by the underlying
supergeometry. In particular, we discuss the harmonic analysis on supergroups
and supercosets and point out the subtleties of Lie superalgebra representation
theory that are responsible for the emergence of logarithmic representations.
Furthermore, special types of marginal deformations of supergroup WZW models
are studied which only exist if the Killing form is vanishing. We show how
exact expressions for anomalous dimensions of boundary fields can be derived
using quasi-abelian perturbation theory. Finally, the knowledge of the exact
spectrum is used to motivate a duality between the OSP(4|2) symmetric
Gross-Neveu model and the S(3|2) supersphere sigma-model.
