In evolutionary genomics, researchers have taken an interest in identifying in the genomes substitutions that subtend convergent phenotypic adaptations. This is a difficult question to address, because genomes contain billions of sites, many of which have substituted in the lineages where the adaptations took place, and yet are not linked to them. Those extra substitutions may be linked to other adaptations, may be neutral, or may be linked to mutational biases. Furthermore, one can think of various ways of defining substitutions of interest, and various methods that match those definitions have been used, resulting in different sets of candidate substitutions. In this manuscript we first clarify how adaptation to convergent phenotypic evolution can manifest itself in coding sequences. Second, we review methods that have been proposed to detect convergent adaptive evolution 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 confounding factor.