  Following Arthur's study of the representations of the orthogonal and
symplectic groups, we prove many cases of both the local and global Arthur
conjectures for tempered representations of the unitary group. This completes
the proof of Arthur's description of the discrete series representations of the
quasi-split $p$-adic unitary group, and Arthur's description of the tempered
discrete automorphic representations of the unitary group, satisfying certain
technical conditions.
