We will discuss the Tukey order, focusing on ultrafilters on the natural numbers. After providing some context on Isbell’s classical problem — asking for the number of equivalence classes in the Tukey order of ultrafilters — I will outline results that establish the consistent non-existence of basically generated ultrafilters, as well as the consistency of all ultrafilters having maximal Tukey type. This is joint work with Jindrich Zapletal.