Human cortical development during the third trimester is characterised by macro- and microstructural changes which are reflected in alterations in diffusion MRI (dMRI) measures, with significant decreases in cortical mean diffusivity (MD) and fractional anisotropy (FA). This has been interpreted as reflecting increased cellular density and dendritic arborisation. However, the fall in FA stops abruptly at 38 weeks post-menstrual age (PMA), and then tends to plateau, while MD continues to fall, suggesting a more complex picture and raising the hypothesis that after this age development is dominated by continuing increase in neural and organelle density rather than alterations in the geometry of dendritic trees. To test this, we used neurite orientation dispersion and density imaging (NODDI), acquiring multi-shell, high angular resolution dMRI and measures of cortical volume and mean curvature in 99 preterm infants scanned between 25 and 47 weeks PMA. We predicted that increased neurite and organelle density would be reflected in increases in neurite density index (NDI), while a relatively unchanging geometrical structure would be associated with constant orientation dispersion index (ODI). As dendritic arborisation is likely to be one of the drivers of gyrification, we also predicted that measures of cortical volume and curvature would correlate with ODI and show slower growth after 38 weeks. We observed a decrease of MD throughout the period, while cortical FA decreased from 25 to 38 weeks PMA and then increased. ODI increased up to 38 weeks and then plateaued, while NDI rose after 38 weeks. The evolution of ODI correlated with cortical volume and curvature. Regional analysis of cortical microstructure revealed a heterogenous pattern with increases in FA and NDI after 38 weeks confined to primary motor and sensory regions. These results support the interpretation that cortical development between 25 and 38 weeks PMA shows a predominant increase in dendritic arborisation and neurite growth, while between 38 and 47 weeks PMA it is dominated by increasing cellular and organelle density.