The stress field in composite elastic media often contains singularities, in particular at the intersections of interfaces with boundaries. These singularities yield 'infinite' values of the stress, leading in practice to damage such as delamination or fracture. This book describes two new methods of computing the eigenvalues and eigenvectors of singularities, leading to a full description of their structure. The necessary results from singularity theory are explained, and many examples are included. The book also gives several important applications in engineering, such as boundary layers in composite plates and slightly perturbed corners.