Finding correspondence between non-rigid 3D shapes is a challenging task underlying numerous problems in computer graphics and vision. In this talk, I will present our recent line of works on deformable partial shape correspondence in the spectral domain. I will first introduce Partial Functional Maps (PFM), showing how to robustly formulate the shape correspondence problem under missing geometry with the language of functional maps. We use perturbation analysis to show how removal of shape parts changes the Laplace-Beltrami eigenfunctions, and exploit it as a prior on the spectral representation of the correspondence. I will show further extensions to deal with the presence of clutter (deformable object-in-clutter) and multiple pieces (non-rigid puzzles). In the second part of the talk, I will show how to formulate the problem completely in the spectral domain, avoiding the cumbersome alternating optimization used in the previous approaches. This allows matching shapes with constant complexity independent of the number of points, and yields state-of-the-art results on challenging correspondence benchmarks in the presence of partiality and topological noise.