Introduction. In this paper we consider the current status of a number of open questions concerning the structural organization of classes of Turing degrees below 0′, the degree of the Halting Problem. We denote the set of all such degrees by D(≤0_).

The Ershov hierarchy arranges these degrees into different levels which are determined by a quantitative characteristic of the complexity of algorithmic recognition of the sets composing these degrees.

The finite level n, n ≥ 1, of the Ershov hierarchy constitutes n-c.e. sets which can be presented in a canonical form as

A (Turing) degree a is called an n-c.e. degree if it contains an n- c.e. set, and it is called a properly n-c.e. degree if it contains an n-c.e. set but no (n – 1)-c.e. sets. We denote by Dn the set of all n- c.e. degrees. R denotes the set of c.e. degrees.

Degrees containing sets from different levels of the Ershov hierarchy, in particular the c.e. degrees, are the most important representatives of D(≤ 0′). Investigations of these degree structures pursued in last two-three decades show that the c.e. degrees and the degrees from finite levels of the Ershov hierarchy have similar properties in many respects.