this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25

We prove that the isomorphism problem for torsion-free Abelian groups is as complicated as any isomorphism problem could be in terms of the analytical hierarchy, namely Σ 1 1 complete.

In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set