  We investigate what it means for a (Hausdorff, second-countable) topological
group to be computable. We compare several potential definitions in the
literature. We relate these notions with the well-established definitions of
effective presentability for discrete and profinite groups, and compare these
results with similar results in computable topology. Most of these definitions
can be separated by counter-examples. Remarkably, we prove that two such
definitions are equivalent for locally compact Polish and abelian Polish
groups. More specifically, we prove that in these broad classes of groups,
every computable topological group admits a right-c.e.~(upper semi-computable)
presentation with a left-invariant metric, and a computable dense sequence of
points. In the locally compact case, we also show that if the group is
additionally effectively locally compact, then we can produce an effectively
proper left-invariant metric.
