  We study Steinberg algebras constructed from ample Hausdorff groupoids over
commutative integral domains with identity. We reconstruct (graded) groupoids
from (graded) Steinberg algebras and use this to characterise when there is a
diagonal-preserving (graded) isomorphism between two (graded) Steinberg
algebras. We apply this characterisation to groupoids of directed graphs in
order to study diagonal-preserving (graded) isomorphisms of Leavitt path
algebras and graph $C^*$-algebras.
