The level structures of the neutron-rich {sup 104}Mo and {sup 108}Mo nuclei have been investigated by observing prompt {gamma} rays emitted in the spontaneous fission of {sup 248}Cm with the EUROGAM spectrometer. Levels with spins up to 12{h_bar} have been observed and {gamma} branching obtained. The data can be satisfactorily described when {sup 104,108}Mo are considered as axially symmetric nuclei: in {sup 104}Mo, rotational bands based on the ground state, the one-phonon and the two-phonon {gamma}-vibrational states and a quasiparticle state have been observed, whereas in {sup 108}Mo the information is limited to the yrast band and the one phonon {gamma} band. {copyright} {ital 1996 The American Physical Society.}