  We prove that for any constant $K>0$ there exists a separable group equipped
with a complete bi-invariant metric bounded by $K$, isometric to the Urysohn
sphere of diameter $K$, that is of `almost-universal disposition'. It is thus
an object in the category of separable groups with bi-invariant metric
analogous in its properties to the Gurarij space from the category of separable
Banach spaces. We show that this group contains an isometric copy of any
separable group equipped with bi-invariant metric bounded by $K$. As a
consequence, we get that it is a universal Polish group admitting compatible
bi-invariant metric, resp. universal second countable SIN group. Moreover, the
almost-universal disposition shows that the automorphism group of this group is
rich and it characterizes the group uniquely up to isometric isomorphism. We
also show that this group is in a certain sense generic in the class of
separable group with bi-invariant metric (bounded by $K$).
  On the other hand, we prove there is no metrically universal separable group
with bi-invariant metric when there is no restriction on diameter. The same is
true for separable locally compact groups with bi-invariant metric.
  Assuming the generalized continuum hypothesis, we prove that there exists a
metrically universal (unbounded) group of density $\kappa$ with bi-invariant
metric for any uncountable cardinal $\kappa$. We moreover deduce that under GCH
there is a universal SIN group of weight $\kappa$ for any infinite cardinal
$\kappa$.
