The failure of the integral Hodge conjecture has been known since the famous counterexamples of Atiyah and Hirzebruch. Currently, several methods exist for producing other counterexamples. For instance, a seminal result of Colliot-Thélène and Voisin relates the failure of the integral Hodge conjecture in degree 4 to degree 3 unramified cohomology. After giving a quick overview of this method, I will discuss how it can be used to obtain new counterexamples to the integral Hodge conjecture.