  We study invariant and bi-invariant metrics on groups focusing on finite
groups $G$. We show that non-equivalent (bi) invariant metrics on $G$ are in
1-1 correspondence with unitary symmetric (conjugate) partitions on $G$. To
every metric group $(G,d)$ we associate to it the symmetry group and the
weighted graph of distances. Using these objects we can classify all
equivalence classes of invariant and bi-invariant metrics for small groups. We
then study the number of non-equivalent invariant and bi-invariant metrics on
$G$. We give an expression for the number of such metrics in terms of Bell
numbers, with closed expressions for certain groups such as abelian, dihedral,
quasidihedral and dicyclic groups. We then characterize all the groups (finite
or not) in which every invariant metric is also bi-invariant. We give the
number of non-equivalent invariant and bi-invariant metrics for all the groups
of order up to 32.
