This chapter describes the adaptation and application of kernel methods for speech processing. It is divided into two sections dealing with speaker verification and isolated-word speech recognition applications. Significant advances in kernel methods have been realised in the field of speaker verification, particularly relating to the direct scoring of variable-length speech utterances by sequence kernel SVMs. The improvements are so substantial that most state-of-the-art speaker recognition systems now incorporate SVMs. We describe the architecture of some of these sequence kernels. Speech recognition presents additional challenges to kernel methods and their application in this area is not as straightforward as for speaker verification. We describe a sequence kernel that uses dynamic time warping to capture temporal information within the kernel directly. The formulation also extends the standard dynamic time-warping algorithm by enabling the dynamic alignment to be computed in a high-dimensional space induced by a kernel function. This kernel is shown to work well in an application for recognising low-intelligibility speech of severely dysarthric individuals.