Complex geometric objects have gained much importance in many different application fields such as medicine, computer aided design or engineering. Modern sensor technologies produce large amounts of 3D (or higher dimensional) data, that need to be analyzed and processed automatically. Methods to compare, recognize and  process shape (2D surfaces or 3D solid objects) are essential ingredients to achieve this goal. This talk will give an overview on different applications of spectral methods in shape analysis. I will demonstrate how the eigenvalues and eigenfunctions of the Laplace-Beltrami operator yield powerful tools to describe and analyze shape. Due to their isometry invariance they are optimally suited to deal with non-rigid shapes often found in nature, such as a body in different postures. The normed beginning sequence of the spectrum can be used as a global signature  for shape matching and database retrieval, while the eigenfunctions and their topological analysis, employing the Morse-Smale complex, persistence diagram or the Reeb graph, can be applied for shape registration, segmentation and local shape analysis. Examples of applications such as database retrieval of near isometric shapes, statistical shape analysis of subcortical structures and hierarchical segmentation of articulated shapes will be presented.