Geometry Processing is a field that aims at providing better tools for the analysis and manipulation of geometric shapes (typically 3d point clouds or triangle meshes). In this talk I will present some recent work, which shows that many standard operations in Geometry Processing can be naturally viewed as linear transformations between functional spaces defined on the shapes, rather than manipulations of vertices or triangles. In particular, I will describe a functional representation of maps (correspondences) between pairs of 3D shapes that generalizes the standard notion of a map to include correspondences that are not necessarily point-to-point, and leads to efficient inference (shape matching). I will also discuss how the operator point of view leads to better map analysis and visualization. Finally, I will describe how the design and analysis of tangent vector fields can be naturally viewed through the lens of linear functional operators, leading to more efficient algorithms and novel applications.