These are the elaborated notes of two talks given at the Summer School in Göttingen on Higher-Dimensional Geometry over Finite Fields. We study the De Rham cohomology of smooth and proper varieties over fields of positive characteristic in case that the Hodge spectral sequence degenerates. The De Rham cohomology carries the structure of a so-called F-zip. We explain two classifications of F-zips, one stems from representation theory of algebras and the other one uses algebraic groups and their compactifications. We show how this second classification can be extended if the De Rham cohomology is endowed with a symplectic or a symmetric pairing. Throughout we illustrate the theory via the examples of (polarized) abelian varieties and (polarized) K3-surfaces.