  We investigate the fundamental concept of a closed quantum subgroup of a
locally compact quantum group. Two definitions - one due to S.Vaes and one due
to S.L.Woronowicz - are analyzed and relations between them discussed. Among
many reformulations we prove that the former definition can be phrased in terms
of quasi-equivalence of representations of quantum groups while the latter can
be related to an old definition of Podle\'s from the theory of compact quantum
groups. The cases of classical groups, duals of classical groups, compact and
discrete quantum groups are singled out and equivalence of the two definitions
is proved in the relevant context. A deep relationship with the quantum group
generalization of Herz restriction theorem from classical harmonic analysis is
also established, in particular, in the course of our analysis we give a new
proof of Herz restriction theorem.
