A real number x is called Δ20 if its binary expansion corresponds to a Δ20-set of natural numbers. Such reals are just the limits of computable sequences of rational numbers and hence also called computably approximable. Depending on how fast the sequences converge, Δ20-reals have different levels of effectiveness. This leads to various hierarchies of Δ20 reals. In this survey paper we summarize several recent developments related to such kind of hierarchies shown by the author and his collaborators. ...