  Nonlinear sigma models with non-compact target space and non-amen-able
symmetry group were introduced long ago in the study of disordered electron
systems. They also occur in dimensionally reduced quantum gravity; recently
they have been considered in the context of the AdS/CFT correspondence. These
models show spontaneous symmetry breaking in {\it any} dimension, even one and
two (superficially in contradiction with the Mermin-Wagner theorem) as a
consequence of the non-amenability of their symmetry group. The low-dimensional
models show other peculiarities: invariant observables remain dependent on
boundary conditions in the thermodynamic limit and the Osterwalder-Schrader
reconstruction yields a non-separable Hilbert space. The ground state space,
however, under quite general conditions, carries a unique unitary and
continuous representation. The existence of a continuum limit in 2D is an open
question: while the perturbative Renormalization Group suggests triviality,
other arguments hint at the existence of a conformally invariant continuum
limit at least for suitable observables.
  This talk gives an overview of the work done during the last several years in
collaboration first of all with Max Niedermaier, some of it also with Peter
Weisz and Tony Duncan.
