It is now generally accepted that EEG is the only reliable way to accurately detect newborn seizures and, as such, prolonged EEG monitoring is increasingly being adopted in neonatal intensive care units. Long EEG recordings may last from several hours to a few days. With neurophysiologists not always available to review the EEG during unsociable hours, there is a pressing need to develop a reliable and robust automatic seizure detection method—a computer algorithm that can take the EEG signal, process it, and output information that supports clinical decision making. In this study, we review existing algorithms based on how the relevant seizure information is exploited. We start with commonly used methods to extract signatures from seizure signals that range from those that mimic the clinical neurophysiologist to those that exploit mathematical models of neonatal EEG generation. Commonly used classification methods are reviewed that are based on a set of rules and thresholds that are either heuristically tuned or automatically derived from the data. These are followed by techniques to use information about spatiotemporal seizure context. The usual errors in system design and validation are discussed. Current clinical decision support tools that have met regulatory requirements and are available to detect neonatal seizures are reviewed with progress and the outstanding challenges are outlined. This review discusses the current state of the art regarding automatic detection of neonatal seizures.