  This paper is the first version of a project of classifying all superelliptic
curves of genus $g \leq 48$ according to their automorphism group. We determine
the parametric equations in each family, the corresponding signature of the
group, the dimension of the family, and the inclussion among such families. At
a later stage it will be determined the decomposition of the Jacobians and each
locus in the moduli spaces of curves.
