Abstract: We introduce frame fields, which are a non-orthogonal and non-unit-length generalization of cross fields. Frame fields represent smoothly varying linear transformations on tangent spaces of a surface. We propose an algorithm to create discrete, dense frame fields that satisfy a sparse set of constraints. By computing a surface deformation that warps a frame field into a cross field, we generalize existing quadrangulation algorithms to generate anisotropic and non-uniform quad meshes whose elements shapes match the frame field. With this, our framework enables users to control not only the alignment but also the density and anisotropy of the elements' distribution, resulting in high-quality adaptive quad meshing.