  We investigate the descriptive complexity of the set of models of first-order
theories. Using classical results of Knight and Solovay, we give a sharp
condition for complete theories to have a $\boldsymbol\Pi_\omega^0$-complete
set of models. We also give sharp conditions for theories to have a
$\boldsymbol\Pi^0_n$-complete set of models. Finally, we determine the Turing
degrees needed to witness the completeness.
