In evolutionary genomics, researchers have taken an interest in identifying substitutions that subtend convergent phenotypic adaptations. This is a difficult question that requires distinguishing foreground convergent substitutions that are involved in the convergent phenotype from background convergent substitutions. Those may be linked to other adaptations, may be neutral or may be the consequence of mutational biases. Furthermore, there is no generally accepted definition of convergent substitutions. Various methods that use different definitions have been proposed in the literature, resulting in different sets of candidate foreground convergent substitutions. In this article, we first describe the processes that can generate foreground convergent substitutions in coding sequences, separating adaptive from non-adaptive processes. Second, we review methods that have been proposed to detect foreground convergent substitutions in coding sequences and expose the assumptions that underlie them. Finally, we examine their power on simulations of convergent changes—including in the presence of a change in the efficacy of selection—and on empirical alignments.