  We study infinite superelliptic curves as translation surfaces and explore
their Veech groups. These objects are branched covering of the complex plane
with branching over infinitely many points. We provide a criterion for
isomorphism between a special family of infinite superelliptic curves. We show
geometric descriptions of saddle connections and holonomy vectors on these
infinite superelliptic curves. We prove that the Veech group of an infinite
superelliptic curve are all the matrices arising from the differential of the
affine mappings $\mathbb{C}$ to itself, permuting the branched points. We
obtain necessary and sufficient conditions to guarantee that the Veech group of
an infinite superelliptic curve is uncountable. We establish a trichotomy on
the holonomy vector set and from it, we give a precise characterization of some
countable groups that can appear as Veech group of an infinite superelliptic
curve. We also construct and study several examples of interesting infinite
superelliptic curves illustrating our results.
