The $(2,3)$-generation of the finite unitary groups

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