  In a first result, we describe all finitely generated factorial algebras over
an algebraically closed field of characteristic zero that come with an
effective multigrading of complexity one by means of generators and relations.
This enables us to construct systematically varieties with free divisor class
group and a complexity one torus action via their Cox rings. For the Fano
varieties of this type that have a free divisor class group of rank one, we
provide explicit bounds for the number of possible deformation types depending
on the dimension and the index of the Picard group in the divisor class group.
As a consequence, one can produce classification lists for fixed dimension and
Picard index. We carry this out expemplarily in the following cases. There are
15 non-toric surfaces with Picard index at most six. Moreover, there are 116
non-toric threefolds with Picard index at most two; nine of them are locally
factorial, i.e. of Picard index one, and among these one is smooth, six have
canonical singularities and two have non-canonical singularities. Finally,
there are 67 non-toric locally factorial fourfolds and two one-dimensional
families of non-toric locally factorial fourfolds. In all cases, we list the
Cox rings explicitly.
