This talk will be about the description of anomalies in quantum field theory through boundary states of certain field theories in one dimensions higher.
I will give a gentle introduction to the mathematical formulation of this approach developed by Freed and Monnier and explain its relation to more classical approaches.
I will discuss two concrete examples in detail, namely the parity anomaly of fermionic gauge theories in odd spacetime dimensions and 't Hooft anomalies of discrete gauge theories.
Time permitting I will mention some open questions related to smooth families of field theories. The talk is based on joint work with R. J. Szabo.