The translational and rotational motions of water and dimethyl sulfoxide, [DMSO, (CH ₃) ₂SO] have been investigated using quasi-elastic neutron scattering. Water-DMSO mixtures at five DMSO mole fractions, χ _DMSO, ranging from 0 to 0.75, were measured. Hydrogen-deuterium substitution was used to extract independently the water proton dynamics (d-DMSO-H ₂O), the DMSO methyl proton dynamics (h-DMSO-D ₂O) and to obtain background corrections (d-DMSO-D ₂O). The translational diffusion of water slows down significantly compared to bulk water at all χ _DMSO>0. The rotational time constant for water exhibits a maximum at χ _DMSO=0.33 that corresponds to the observed maximum of the viscosity of the mixture. Data for DMSO can be analyzed in terms of a relatively slow tumbling of the molecule about its center-of-mass in conjunction with random translational diffusion. The rotational time constant for this motion exhibits some dependence on χ _DMSO, while the translational diffusion constant shows no clear variation for χ _DMSO>0. The results presented reinforce the idea that due to the stronger associative nature of DMSO, DMSO-water aggregates are formed over the whole composition range, disturbing the tetrahedral natural arrangement of the water molecules. As a consequence adding DMSO to water causes a drastic slowing down of the dynamics of the water molecule, and vice versa.