In this chapter, we consider the computability-theoretic complexities of the theories of various metric structures. In the first section, we consider the complexity of the diagram of a metric structure and establish results for first-order formulas as well as for infinitary formulas. We then study the complexity of the theory of a structure. As alluded to in Chap.  2 , there are two perspectives on the theory of a metric structure, and these two perspectives yield slightly different effective results. Afterwards, we consider a particular kind of structure arising in model theory, the finitely generic structures, and establish upper bounds on the complexities of their theories. The final section studies the universal theory of the hyperfinite II \(_1\) factor, whose undecidability is intimately connected with a recent result of quantum complexity theory known as MIP \(^*=\ \) RE.