Techniques which utilise a signal's Higher Order Statistics (HOS) can reveal information about non-Gaussian signals and nonlinearities which cannot be obtained using conventional (second-order) techniques. This information may be useful in speech processing because it may provide clues about how to construct new models of speech production which are better than existing models. There has been a recent surge of interest in the application of HOS techniques to speech processing, but this has been handicapped by a lack of understanding of what the HOS properties of speech signals are. Without this understanding the HOS information which is in speech signals can not be efficiently utilised. This thesis describes an investigation into the use of HOS techniques, in particular the third-order frequency domain measure called the bispectrum, to speech signals. Several issues relating to bispectral speech analysis are addressed, including nonlinearity detection, pitch-synchronous analysis, estimation criteria and stationarity. A flaw is identified in an existing algorithm for detecting quadratic nonlinearities, and a new detector is proposed which has better statistical properties. In addition, a new algorithm is developed for estimating the normalised bispectrum of signals contaminated by transient noise. Finally the tools developed in the study are applied to a specially constructed database of continuant speech sounds. The results are consistent with the hypothesis that speech signals do not exhibit quadratic nonlinearity.