  An ultragraph gives rise to a labelled graph with some particular properties.
In this paper we describe the algebras associated to such labelled graphs as
groupoid algebras. More precisely, we show that the known groupoid algebra
realization of ultragraph C*-algebras is only valid for ultragraphs for which
the range of each edge is finite, and we extend this realization to any
ultragraph (including ultragraphs with sinks). Using our machinery, we
characterize the shift space associated to an ultragraph as the tight spectrum
of the inverse semigroup associated to the ultragraph (viewed as a labelled
graph). Furthermore, in the purely algebraic setting, we show that the
algebraic partial action used to describe an ultragraph Leavitt path algebra as
a partial skew group ring is equivalent to the dual of a topological partial
action, and we use this to describe ultragraph Leavitt path algebras as
Steinberg algebras. Finally, we prove generalized uniqueness theorems for both
ultragraph C*-algebras and ultragraph Leavitt path algebras and characterize
their abelian core subalgebras.
