Cerebral cortical folding becomes dramatically more complex in the fetal brain during the 3rd trimester of gestation; the process continues in a similar fashion in children who are born prematurely. To quantify this morphological development, it is necessary to extract the interface between gray matter and white matter, which is particularly challenging due to changing tissue contrast during brain maturation. We employed the well-established CIVET pipeline to extract this cortical surface, with point correspondence across subjects, using a surface-based spherical registration. We then developed a variant of the pipeline, called NEOCIVET, that quantified cortical folding using mean curvature and sulcal depth while addressing the well-known problems of poor and temporally-varying gray/white contrast as well as motion artifact in neonatal MRI. NEOCIVET includes: i) a tissue classification technique that analyzed multi-atlas texture patches using the nonlocal mean estimator and subsequently applied a label fusion approach based on a joint probability between templates, ii) neonatal template construction based on age-specific sub-groups, and iii) masking of non-interesting structures using label-fusion approaches. These techniques replaced modules that might be suboptimal for regional analysis of poor-contrast neonatal cortex. The proposed segmentation method showed more accurate results in subjects with various ages and with various degrees of motion compared to state-of-the-art methods. In the analysis of 158 preterm-born neonates, many with multiple scans (n=231; 26-40weeks postmenstrual age at scan), NEOCIVET identified increases in cortical folding over time in numerous cortical regions (mean curvature: +0.003/week; sulcal depth: +0.04mm/week) while folding did not change in major sulci that are known to develop early (corrected p<0.05). The proposed pipeline successfully mapped cortical structural development, supporting current models of cerebral morphogenesis, and furthermore, revealed impairment of cortical folding in extremely preterm newborns relative to relatively late preterm newborns, demonstrating its potential to provide biomarkers of prematurity-related developmental outcome.