Building on the concepts of differential geometry and the discrete counterparts introduced in Chapter 3, in this chapter we present mesh smoothing. On an abstract level, mesh smoothing is concerned with the design and computation of smooth functions f : S → IRd on a triangle mesh. Due to this very general formulation, mesh smoothing is a fundamental tool in geometry processing. The function f can flexibly be chosen to describe, for instance, vertex positions, texture coordinates, or vertex displacements, such that the techniques introduced in this chapter can be used for mesh parameterization (Chapter 5), isotropic remeshing (Chapter 6), hole filling (Chapter 8), and mesh deformation (Chapter 9).