Abstract: Ru-116 and Ru-118 have been studied via beta-delayed gamma-ray spectroscopy of nuclei produced in fragmentation reactions at the Radioactive Ion-Beam Factory (RIBF) facility. Level schemes with positive-parity states up to spin J = 6 have been constructed. The results have been discussed in terms of the interacting boson model, the algebraic collective model, and total Routhian surfaces. We conclude that the very neutron-rich nuclei still show many features associated with triaxial gamma-soft nuclei, represented by the O(6) symmetry, but are approaching a spherical structure, the U(5) symmetry, with increasing neutron number towards the N = 82 shell closure. In Ru-118, hints of a shape transition in the ground state have been observed.