The structural and optical properties of InGaN/GaN nanowire heterostructures grown by plasma-assisted molecular beam epitaxy have been studied using a combination of transmission electron microscopy, electron tomography and photoluminescence spectroscopy. It is found that, depending on In content, the strain relaxation of InGaN may be elastic or plastic. Elastic relaxation results in a pronounced radial In content gradient. Plastic relaxation is associated with the formation of misfit dislocations at the InGaN/GaN interface or with cracks in the InGaN nanowire section. In all cases, a GaN shell was formed around the InGaN core, which is assigned to differences in In and Ga diffusion mean free paths.