  We review the concept of the (anomalous) Poisson-Lie symmetry in a way that
emphasises the notion of Poisson-Lie Hamiltonian. The language that we develop
turns out to be very useful for several applications: we prove that the left
and the right actions of a group $G$ on its twisted Heisenberg double
$(D,\kappa)$ realize the (anomalous) Poisson-Lie symmetries and we explain in a
very transparent way the concept of the Poisson-Lie subsymmetry and that of
Poisson-Lie symplectic reduction. Under some additional conditions, we
construct also a non-anomalous moment map corresponding to a sort of
quasi-adjoint action of $G$ on $(D,\kappa)$. The absence of the anomaly of this
"quasi-adjoint" moment map permits to perform the gauging of deformed WZW
models.
