  We study boundary conditions of topological sigma models with the goal of
generalizing the concepts of anomalous symmetry and symmetry protected
topological order. We find a version of 't Hooft's anomaly matching conditions
on the renormalization group flow of boundaries of invertible topological sigma
models and discuss several examples of anomalous boundary theories. We also
comment on bulk topological transitions in dynamical sigma models and argue
that one can, with care, use topological data to draw sigma model phase
diagrams.
