Developing techniques to monitor cerebral function is becoming increasingly recognised as a way forward to detect and understand brain injury in patients. Diffuse optical tomography (DOT) is a promising imaging technique that is non-invasive, portable and relatively simple to operate. In this thesis, I describe the development and design of DOT headgear for neonates and subsequently apply this in neonatal studies of cerebral function. Firstly, the DOT headgear was applied to healthy neonates to investigate the early stages of vocal specialisation. In this functional activation study, the cerebral haemodynamic changes observed in response to vocal and non-vocal stimuli are presented in Chapter 6. In recent years, the study of spontaneous cerebral activity that gives rise to resting state networks (RSNs) is becoming widely popular. These studies rely on infants to remain asleep to minimise subject motion and optimise data quality. Using DOT combined with electroencephalography (EEG), the effect of sleep state on RSNs was studied for the first time in healthy neonates with compelling results that are presented in Chapter 7. In the context of sleep state and RSNs, combined DOT-EEG was also applied to neurologically compromised patients in the neonatal intensive care unit (NICU). The results of this case-based study are presented in Chapter 8. These studies demonstrate the potential clinical use of DOT in neonates to evaluate cerebral function. In completion of this thesis, the implications of the results and findings from the studies are discussed and future directions explored.