  We prove a Kuranishi-type theorem for deformations of complex structures on
ALE K\"ahler surfaces. This is used to prove that for any scalar-flat K\"ahler
ALE surface, all small deformations of complex structure also admit scalar-flat
K\"ahler ALE metrics. A local moduli space of scalar-flat K\"ahler ALE metrics
is then constructed, which is shown to be universal up to small diffeomorphisms
(that is, diffeomorphisms which are close to the identity in a suitable sense).
A formula for the dimension of the local moduli space is proved in the case of
a scalar-flat K\"ahler ALE surface which deforms to a minimal resolution of
$\mathbb{C}^2/\Gamma$, where $\Gamma$ is a finite subgroup of ${\rm{U}}(2)$
without complex reflections.
