  Let $\rho$ be the $p$-adic Galois representation attached to a cuspidal,
regular algebraic automorphic representation of $\mathrm{GL}_n$ of unitary
type. Under very mild hypotheses on $\rho$, we prove the vanishing of the
(Bloch--Kato) adjoint Selmer group of $\rho$. We obtain definitive results for
the adjoint Selmer groups associated to non-CM Hilbert modular forms and
elliptic curves over totally real fields.
