This article presents new operational convergence criteria for the Gibbs sampler (GS) and related procedures that are useful for determining whether they not only have converged but also have converged to provide reliable results. Three GS convergence criteria are presented and applied: the difference convergence criterion (DC2), the ratio convergence criterion (RC2), and the anchored ratio convergence criterion (ARC2). Their uses and properties are discussed and examples are analyzed to illustrate their application.