  We prove a uniqueness theorem and give a characterization of simplicity for
Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a
uniqueness theorem and give a characterization of simplicity for the C*-algebra
associated to non-Hausdorff \'etale groupoids. Then we show how our results
apply in the setting of tight representations of inverse semigroups, groups
acting on graphs, and self-similar actions. In particular, we show that
C*-algebra and the complex Steinberg algebra of the self-similar action of the
Grigorchuk group are simple but the Steinberg algebra with coefficients in
$\mathbb{Z}_2$ is not simple.
