  What does it mean for a boundary condition to be symmetric with respect to a
non-invertible global symmetry? We discuss two possible definitions in 1+1d. On
the one hand, we call a boundary weakly symmetric if the symmetry defects can
terminate topologically on it, leading to conserved operators for the
Hamiltonian on an interval (in the open string channel). On the other hand, we
call a boundary strongly symmetric if the corresponding boundary state is an
eigenstate of the symmetry operators (in the closed string channel). These two
notions of symmetric boundaries are equivalent for invertible symmetries, but
bifurcate for non-invertible symmetries. We discuss the relation to anomalies,
where we observe that it is sometimes possible to gauge a non-invertible
symmetry in a generalized sense even though it is incompatible with a trivially
gapped phase. The analysis of symmetric boundaries further leads to constraints
on bulk and boundary renormalization group flows. In 2+1d, we study the action
of non-invertible condensation defects on the boundaries of $U(1)$ gauge theory
and several TQFTs. Starting from the Dirichlet boundary of free Maxwell theory,
the non-invertible symmetries generate infinitely many boundary conditions that
are neither Dirichlet nor Neumann.
