  Recently, general methods of bosonization beyond 1+1 dimensions have been
developed. In this article, we review these bosonizations and extend them to
the case with boundary conditions. In particular, we study the case when the
bulk theory is a $G$-symmetry protected topological phase and the boundary
theory has a $G$ 't Hooft anomaly. We discuss how, when the anomaly is not
realizable in a bosonic system, the $G$ symmetry algebra becomes modified in
the bosonization of the anomalous theory. This gives us a useful tool for
understanding anomalies of fermionic systems, since there is no way to compute
a boundary gauge variation of the anomaly polynomial, as one does for anomalies
of bosonic systems. We take the chiral anomalies in 1+1D and the parity/time
reversal anomalies in 2+1D as case studies. We also provide a derivation of new
constraints in SPT phases with domain defects decorated by $p+ip$
superconductors and Kitaev strings.
