The overall aim of this thesis was to characterize single-channel aEEG/EEG, recorded during the first postnatal days in preterm infants, in relation to brain function and two-year outcome. Study I investigated if aEEG/EEG was associated with neonatal brain injury, inflammation and outcome in 16 very preterm (VPT) infants. The interburst interval (IBI) was prolonged, and aEEG amplitudes were lower in infants with brain injury, and in infants developing handicap. Cord blood TNF-α correlated with IBI. Study II investigated inter-rater agreement of visual burst detection, as compared to automated burst detection based on a non-linear energy operator (NLEO) in an EEG data set from 12 extremely preterm (EPT) and 6 VPT infants. The sensitivity of the NLEO was 64 % and 69 % (EPT and VPT infants, respectively) and the specificity 96 % and 88 %. The algorithm was then modified to further improve the accuracy. Study III investigated if arterial carbon dioxide and plasma glucose is associated with EEG continuity. In 247 sets of samples (PaCO2, plasma glucose, IBI) from 32 EPT infants there was a positive association between PaCO2 and IBI; higher PaCO2 was associated with longer IBI. Corrected for carbon dioxide, plasma glucose had a U-shaped association with IBI in infants with good outcome. Study IV investigated the predictive value of aEEG/EEG in 41 EPT and 8 VPT infants. All VPT infants had good outcome. Predictors of outcome in EPT infants included presence or absence of burst-suppression, continuous activity and cyclicity, median IBI and interburst%. Seizures were associated with neonatal brain damage but not with outcome. Improved preterm brain monitoring may in the future be used for early identification of infants at high risk of brain damage and adverse outcome, which may have implications for direction of care and for early intervention.