  In this paper we prove that the unitary groups $SU_n(q^2)$ are
$(2,3)$-generated for any prime power $q$ and any integer $n\geq 8$. By
previous results this implies that, if $n\geq 3$, the groups $SU_n(q^2)$ and
$PSU_n(q^2)$ are $(2,3)$-generated, except when
$(n,q)\in\{(3,2),(3,3),(3,5),(4,2), (4,3),(5,2)\}$.
