The overall aim of our research is to develop methods for a monitoring system to be used at neonatal intensive care units. When monitoring a baby, a range of different types of background activity needs to be considered. In this work, we have developed a scheme for automatic classification of background EEG activity in newborn babies. EEG from six full-term babies who were displaying a burst suppression pattern while suffering from the after-effects of asphyxia during birth was included along with EEG from 20 full-term healthy newborn babies. The signals from the healthy babies were divided into four behavioural states: active awake, quiet awake, active sleep and quiet sleep. By using a number of features extracted from the EEG together with Fisher's linear discriminant classifier we have managed to achieve 100% correct classification when separating burst suppression EEG from all four healthy EEG types and 93% true positive classification when separating quiet sleep from the other types. The other three sleep stages could not be classified. When the pathological burst suppression pattern was detected, the analysis was taken one step further and the signal was segmented into burst and suppression, allowing clinically relevant parameters such as suppression length and burst suppression ratio to be calculated. The segmentation of the burst suppression EEG works well, with a probability of error around 4%.