Introduction. This paper is a follow-up to the author's paper Five Lectures on Algorithmic Randomness, Downey [6], and covers material not covered in those lectures. In particular, I plan to look at lowness which was the basis of one of my lectures in Nijmegen, and to try to make accessible the decanter method whose roots come from the paper Downey, Hirschfeldt, Nies and Stephan [10], and whose full development was in Nies [23]. The class of K-trivial reals has turned out to be a remarkable “natural” class with really fascinating properties and connections with other areas of mathematics. For instance, K-trivial reals allow us to solve Post's problem “naturally” (well, reasonably naturally), without the use of a priority argument, or indeed without the use of requirements. Also, Kučera and Slaman [17] have used this class to solve a longstanding question in computable model theory, namely given a noncomputable Y in a Scott set S, there exists in S an element X Turing incomparable with Y.

As well I will report on recent work of the author with Peter Cholak and Noam Greenberg [4] on the computably enumerable strongly jump traceable reals which form a proper subclass of the K-trivials.