  This review is dedicated to two-dimensional sigma models with flag manifold
target spaces, which are generalizations of the familiar $CP^{n-1}$ and
Grassmannian models. They naturally arise in the description of continuum
limits of spin chains, and their phase structure is sensitive to the values of
the topological angles, which are determined by the representations of spins in
the chain. Gapless phases can in certain cases be explained by the presence of
discrete 't Hooft anomalies in the continuum theory. We also discuss integrable
flag manifold sigma models, which provide a generalization of the theory of
integrable models with symmetric target spaces. These models, as well as their
deformations, have an alternative equivalent formulation as bosonic Gross-Neveu
models, which proves useful for demonstrating that the deformed geometries are
solutions of the renormalization group (Ricci flow) equations, as well as for
the analysis of anomalies and for describing potential couplings to fermions.
