Awareness under anaesthesia has become an issue of increasing importance over recent years. The analysis of the EEG as a means of determining the level of consciousness during an operation is a very attractive option, since taking measurements from the scalp is logistically straightforward. Considerable research effort has been directed to this problem, and there is a large body of literature arising from related investigations into areas such as the use of the EEG in the diagnosis of neurological conditions, and the applications of signal processing techniques to the analysis of the EEG more generally. The purpose of this report is to review some of the relevant literature and, concentrating on the signal processing issues, make an assessment of the potential for finding a technique for extracting a meaningful and reliable indication of awareness from either stimulated or background EEG.