Quantitative MRI techniques based on morphology and tissue microstructure dependent contrast provide a unique window on brain development in the neonatal period. The dramatic changes in morphology and MRI contrast that occur during this period have the potential to be used to identify normal and abnormal developmental trajectories that predict neurodevelopmental outcome in at risk populations. Here, we review these technologies focussing on two broad categories: gross morphological analysis and tissue microstructure assessment. With respect to morphology, we examine the role of image registration and atlas-based techniques, highlighting the challenges posed by the scale of the anatomical changes and the high incidence of radiologically abnormal scans in the premature infant population. With respect to microstructure, we examine the potential and remaining challenges for using quantitative MRI to dissociate processes of cell proliferation, neuronal maturation, and myelination by combining different signal contrasts. Recent progress from our group in this area is highlighted.