The Tukey order finds its origins in the concept of Moore-Smith convergence in topology and is especially important when restricted to ultrafilters with reverse inclusion. The Tukey order on ultrafilters over $\omega$ was studied intensively by many, but still contains several fundamental unresolved problems. After providing motivation for studying the Tukey order, I will present a recently discovered connection to a parallel study at the realm of measurable cardinals, and explain how different the Tukey order is at that level when compared to the situation on $\omega$.
In the second part of the talk, I will demonstrate how ideas and intuition from ultrafilters over measurable cardinals led to new results at the level of $\omega$ and present an essentially new method of constructing Tukey-top ultrafilters using Diamond-like Principles on $\omega$.