Steady-state diffusional stress relaxation in a solid fibre is considered. The first model presented deals with free wires containing periodic grain boundaries placed under uniaxial loading. The second considers hard, elastic fibres embedded in a plastically deformed matrix. For free wires, creep due to both volume and grain boundary diffusion is considered in such a way that the diffusive fluxes due to both mechanisms are intimately coupled in the solution. Capillarity induced thermal grooves on the wire surface are found to have a negligible effect on the creep rate. For embedded fibres, the stress in the fibre is calculated for an arbitrary plastic strain in the matrix, using Eshelby's theory for elastic inclusions. For fibres aligned parallel with the tensile axis of the composite, the stress produced in the fibre is primarily tensile and may result in significant diffusional creep once the matrix undergoes about 0.1% tensile strain (or less for small radius fibres at high temperatures). This is also the case for free wires of sub-micron radius at small stresses. Fine fibres (such as in eutectic composites. may lose their load-bearing capabilities at high operating temperatures because of the rate of diffusional relaxation.