We show how Abstract Complexity Theory is related to the degrees of unsolvability and develop machinery by which computability theoretic hierarchies with a complexity theoretic flavor can be defined and investigated. This machinery is used to prove results both on hierarchies of Δ 2 0 sets and hierarchies of Δ 2 0 degrees. We prove a near-optimal lower bound on the effectivity of the Low Basis Theorem and a result showing that array computable c.e. degrees are, in some sense, the (...) simplest possible Δ 2 0 degrees. We also examine the growth rates of iterates of m K . Finally, we indicate how complexity theory can be used to analyze notions of genericity intermediate between 1 -genericity and 2 -genericity, and produce a hierarchy of such notions. (shrink)