The paper under review is a reasonably self-contained exposition of the corona theorems for almost periodic functions on \(\mathbb R^n\). Let \(AP^{+}_{\Sigma}(\mathbb R^n)\) denote the Banach algebra of all continuous almost periodic functions on \(\mathbb R^n\) whose Bohr-Fourier spectrum is contained in an additive semigroup \(\Sigma \subset [0,\infty)^n\). The author shows that the maximal ideal space of \(AP^{+}_{\Sigma}(\mathbb R^n)\) may have a nonempty corona and characterizes all the semigroups \(\Sigma\) for which the corona is empty. In addition, the author establishes analogous results for algebras of almost periodic functions with absolutely convergent Fourier series.