In this thesis, wavelets are applied to the analysis of two types of medical signals, namely infant breathing signals and ultrasound images. One-dimensional wavelets are used to quantify amplitude modulation of infant breathing that occurs during quiet sleep. Two-dimensional wavelets are used to develop enhancement techniques in the wavelet domain, tailored to ultrasound images. The development of wavelets is described from the backgrounds of mathematics, signal processing and sub-band coding. One-dimensional wavelets are defined for both continuous and discrete cases. Two-dimensional wavelets are developed for both separable and non-separable classes. The construction of both one- and two-dimensional wavelets is described, and examples of wavelets that are used in subsequent analyses are presented. The analysis of breathing signals provides information for understanding the physiology of breathing. The wavelet domain is shown to isolate frequency characteristics of breathing signals, as well as indicating the temporal position of those characteristics in the signal. Some of the characteristics that were measured were not distinguished in the original signals. The extent of constant frequency components due to amplitude modulation of the principal breathing rate is quantified. Breathing in infants is of particular interest, as it may provide insight into the cause of Sudden Infant Death Syndrome (SIDS), or cot death. Studies of infants who later succumbed to SIDS, or were at high risk for SIDS, and infants at low risk for SIDS were carried out. The infants who later succumbed to SIDS and those at high risk for SIDS showed different characteristics in the wavelet domain compared to infants at low risk for SIDS. As well as implying that there may be a difference in the physiologies of infants at high and low risk for SIDS, this result confirms that wavelets can be used to analyse breathing signals and produce meaningful results. Ultrasound images typically contain artefacts and low contrast between features of interest and often exhibit a noisy background. The aim of any enhancement procedure is to reduce the contributions from noise and artefacts and increase the contrast between the features of interest and the background. A method to objectively measure improvements in image quality is described, although this attempt achieved mixed results. Separable and non-separable two-dimensional wavelets are used with enhancement functions to improve the image quality. Enhancement schemes, including noise reduction and contrast enhancement, are developed. A spatially varying contrast enhancement function is also developed. Noise reduction, in combination with the spatially varying contrast enhancement function, produces an image that reduces the existing artefacts in the image and increases the contrast for the features of interest. Wavelets are shown to be a useful tool for the analysis of infant breathing signals and for improvement of image quality in ultrasound images.