  New interpolation and quasi-interpolation operators of Cl\'ement- and
Scott-Zhang-type are analyzed on anisotropic polygonal and polyhedral meshes.
Since no reference element is available, an appropriate linear mapping to a
reference configuration plays a crucial role. A priori error estimates are
derived respecting the anisotropy of the discretization. Finally, the found
estimates are employed to propose an adaptive mesh refinement based on
bisection which leads to highly anisotropic and adapted discretizations with
general element shapes in two- and three-dimensions.
