In this thesis, we establish a pair of systems of higher order cohomology operations that act on the Ζ_p-ordinary cohomology of spaces that are free of p-torsion. These "pyramids" of operations are generated by the p-divisibility of certain sums of "pseudo" primary cohomology operations that operate on the p-local cohomology of p-torsion-free spaces. The properties of these higher order operations allow us to prove theorems that either generalise or improve (in the sense of decreasing indeterminacy) several results in the literature.