Recent claims that arthropods are monophyletic because all have jaws composed of a five-segmented coxa, that the groundplan of arthropod legs has no less than 11 segments, that crustaceans, chelicerates and insects share a ‘polyramous arthropod leg’, and that the labrum is formed from a pair of legs, are rejected on factual grounds. It is suggested that the earliest arthropod appendages were unsegmented. Putative homologies among mandibulate arthropods are considered. Striking as some of these are, a good case can be made for their convergent evolution, and the concept of the Mandibulata is rejected. Suggested separate ancestries of crustaceans and tracheates are compared. A realistic explanation of radiation from a common arthropod ancestor remains illusory. A polyphyletic concept of arthropod evolution from soft-bodied, segmented, haemocoele-possessing, non-annelid worms is elaborated. The degree of convergence demanded is amply matched by proven examples of the phenomenon. If the earliest arthropods lacked compound eyes, and these were acquired several times, as they have been at least twice in non-arthropods, several otherwise intractable problems are resolved. Sequence comparisons provide a powerful tool for determining relationships but seem powerless to establish whether arthropods are monophyletic, or polyphyletic in the manner envisaged here.