The stationary wavelet transform and its statistical applications are discussed in this paper. The paper reviews the basics of the discrete wavelet transform using a filter notation that is useful for subsequent discussion. A stationary wavelet transform is described, where the coefficient sequences are not decimated at each stage. Two different approaches to the construction of an inverse of the stationary wavelet transform are presented. The paper discusses the application of the stationary wavelet transform as an exploratory statistical method and its potential use in nonparametric regression. A method of local spectral density estimation is developed, extending standard time series ideas such as the periodogram and spectrum to the wavelet context. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy. The paper begins with an introduction to wavelets and their statistical applications, focusing on the stationary wavelet transform. The stationary wavelet transform is discussed, which is an extension of the standard discrete wavelet transform. The stationary wavelet transform is described as a method that fills in the gaps caused by the decimation step in the standard wavelet transform, leading to an over-determined, or redundant, representation of the original data. This has considerable statistical potential, which is explored in the remainder of the paper. Two main examples, one from astronomy and one from veterinary science, are presented in Section 4, and the value of the stationary wavelet transform for exploring these data sets is discussed. The potential uses of the stationary wavelet transform in regression and curve estimation are then considered in Section 5. In order to do this, the paper sets out ways of inverting the stationary wavelet transform, again building on work of Pesquet et al. The stationary wavelet transform has a valuable role in the exploration and spectral analysis of non-stationary time series. In Section 6, the paper gives a heuristic discussion of the way in which wavelets can be used to extend to non-stationary data the notions of periodogram and spectrum that arise from the Fourier analysis of stationary series. This discussion is illustrated by the application of the methods proposed to the two practical examples introduced in Section 4.The stationary wavelet transform and its statistical applications are discussed in this paper. The paper reviews the basics of the discrete wavelet transform using a filter notation that is useful for subsequent discussion. A stationary wavelet transform is described, where the coefficient sequences are not decimated at each stage. Two different approaches to the construction of an inverse of the stationary wavelet transform are presented. The paper discusses the application of the stationary wavelet transform as an exploratory statistical method and its potential use in nonparametric regression. A method of local spectral density estimation is developed, extending standard time series ideas such as the periodogram and spectrum to the wavelet context. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy. The paper begins with an introduction to wavelets and their statistical applications, focusing on the stationary wavelet transform. The stationary wavelet transform is discussed, which is an extension of the standard discrete wavelet transform. The stationary wavelet transform is described as a method that fills in the gaps caused by the decimation step in the standard wavelet transform, leading to an over-determined, or redundant, representation of the original data. This has considerable statistical potential, which is explored in the remainder of the paper. Two main examples, one from astronomy and one from veterinary science, are presented in Section 4, and the value of the stationary wavelet transform for exploring these data sets is discussed. The potential uses of the stationary wavelet transform in regression and curve estimation are then considered in Section 5. In order to do this, the paper sets out ways of inverting the stationary wavelet transform, again building on work of Pesquet et al. The stationary wavelet transform has a valuable role in the exploration and spectral analysis of non-stationary time series. In Section 6, the paper gives a heuristic discussion of the way in which wavelets can be used to extend to non-stationary data the notions of periodogram and spectrum that arise from the Fourier analysis of stationary series. This discussion is illustrated by the application of the methods proposed to the two practical examples introduced in Section 4.