  We classify compact, connected Hamiltonian and quasi-Hamiltonian manifolds of
cohomogeneity one (which is the same as being multiplicity free of rank one).
Here the group acting is a compact connected Lie group (simply connected in the
quasi-Hamiltonian case). This work is a concretization of the more general
classification (arXiv:1612.03843) of multiplicity free manifolds in the special
case of rank one. As a result we obtain numerous new concrete examples of
multiplicity free quasi-Hamiltonian manifolds or, equivalently, Hamiltonian
loop group actions.
