  We study the question of when the algebra of an ample groupoid over a field
is von Neumann regular and, more generally, when the algebra of a graded ample
groupoid is graded von Neumann regular. For fields of characteristic $0$ we are
able to provide a complete characterization. Over fields of positive
characteristic, we merely have some sufficient and some necessary conditions
for regularity. As applications we recover known results on regularity and
graded regularity of Leavitt path algebras and inverse semigroups. We also
prove a number of new results, in particular concerning graded regularity of
algebras of Deaconu-Renault groupoids and Nekrashevych-Exel-Pardo algebras of
self-similar groups.
