A computable structure $\mathcal{A}$ is $\mathbf{x}$-computably categorical for some Turing degree $\mathbf{x}$ if for every computable structure $\mathcal{B}\cong \mathcal{A}$ there is an isomorphism $f:\mathcal{B}\to \mathcal{A}$ with $f{\le }_T\mathbf{x}$. A degree $\mathbf{x}$ is a degree of categoricity if there is a computable structure $\mathcal{A}$ such that $\mathcal{A}$ is $\mathbf{x}$-computably categorical, and for all $\mathbf{y}$, if $\mathcal{A}$ is $\mathbf{y}$-computably categorical, then $\mathbf{x}{\le }_T\mathbf{y}$.

We construct a ${\Sigma }_2^0$ set whose degree is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity.