The degree of categoricity of a computable structure is the least Turing degree that computes an isomorphism between any two computable isomorphic copies of the structure. It provides a robust measure of complexity for computable structures – if A has degree of categoricity d, then modulo d all its isomorphic copies have the same computational properties. Two of the main goals in this area are to obtain a characterization of the Turing degrees that are degrees of categoricity of computable structures and to see whether all degrees of categoricity are strong, i.e., arise from the isomorphisms between two copies of a structure.

In this talk I will introduce the methods and main results in the area and discuss recent results of Csima and me. We obtained a characterization of the strong degrees of categoricity above 0'' as what we call the treeable degrees above 0''.