  A global symmetry of a quantum field theory is said to have an 't Hooft
anomaly if it cannot be promoted to a local symmetry of a gauged theory. In
this paper, we show that the anomaly is also an obstruction to defining
symmetric boundary conditions. This applies to Lorentz symmetries with
gravitational anomalies as well. For theories with perturbative anomalies, we
demonstrate the obstruction by analyzing the Wess-Zumino consistency conditions
and current Ward identities in the presence of a boundary. We then recast the
problem in terms of symmetry defects and find the same conclusions for
anomalies of discrete and orientation-reversing global symmetries, up to the
conjecture that global gravitational anomalies, which may not be associated
with any diffeomorphism symmetry, also forbid the existence of boundary
conditions. This conjecture holds for known gravitational anomalies in $D \le
3$ which allows us to conclude the obstruction result for $D \le 4$.
