Several hypotheses supporting monophyly of the Teleostei on the basis of synapomorphies have been produced over the last 30 years. The concept of Teleostei sensu Patterson (1977) and Patterson and Rosen (1977), with halecomorphs as the sister-group and dagPachycormiformes and dagAspidorhynchiformes at the base, has been questioned recently. A new hypothesis has been suggested (Arratia, 1999, 2000a) with dagPholidophorus as the basal taxon. Whereas the monophyly of Teleostei has been supported by numerous investigations based on morphological evidence of fossil and living forms and on molecular data, the sister-group of Teleostei is still unresolved. Possible sister-groups are the Amiiformes, Lepisosteiformes, dagDapedium, dagPycnodontiformes, dagPachycormiformes, and dagAspidorhynchiformes. Their relative positions in the cladogram changes when different outgroups are used. The large actinopterygian clade comprising the stem-groups of teleosts and the Teleostei (including fossil and extant members) and excluding the Halecomorphi (Amia and relatives) and the Ginglymodi (Lepisosteus and relatives) is formally named Teleosteomorpha. The name Teleostei is reserved here for the apomorphy-based taxon Teleostei that includes the fossil basal teleosts and the Teleocephala (crown-group). The monophyly of Teleostei is supported by one uniquely derived character and numerous homoplasious derived characters. The available information reveals that there is no correlation between the age of the probable sister-groups of Teleostei and their primitiveness. All of them appeared at the same time as the Teleostei (e.g., dagPycnodontiformes) or are younger (e.g., Amiiformes, Lepisosteiformes, dagDapedium, dagPachycormiformes, and dagAspidorhynchiformes). The closest living (Amia or Lepisosteus) and the closest fossil sister-group of teleosts remains unknown.