I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $G$ is a hyperbolic group and $g \in G$ has infinite order, then the quotient group $G/\langle (g^n)^G \rangle$ is hyperbolic for all sufficiently large $n > 0$.
I have been searching for references, but without success. Can anyone help?.
$\mathbf{Edit}$: After looking at the references in the answer by Mikael de la Salle, I see that I did not state this result correctly. Rather than the statement being for all sufficiently large $n>0$, it should be that there exists and $N>0$ such that $G/\langle (g^{nN})^G \rangle$ is hyperbolic for all $n > 0$. The result stated applies only to non-elementary hyperbolic groups, but for an elementary hyperbolic group this quotient is finite, and so it remains correct.