  This paper is a new important step towards the complete classification of the
finite simple groups which are $(2, 3)$-generated. In fact, we prove that the
symplectic groups $Sp_{2n}(q)$ are $(2,3)$-generated for all $n\geq 4$. Because
of the existing literature, this result implies that the groups $PSp_{2n}(q)$
are $(2,3)$-generated for all $n\geq 2$, with the exception of $PSp_4(2^f)$ and
$PSp_4(3^f)$.
