The Palaeoptera is the group of primitive (basal) winged insects which do not fold the wings back over the abdomen, as Neoptera do. Most of them are extinct; the living ones are the Ephemeroptera (mayflies) and Odonata (dragonflies and damselflies).
The Palaeoptera may not be monophyletic: the issue is not yet settled. On the other hand, the Neoptera definitely are monophyletic.
The absence of wing-folding does not necessarily mean the Palaeoptera form a natural group. They may just be a group of insects that "are not Neoptera", in other words, a wastebasket taxon. As it stands, the relationship of the two living Paleopteran groups to the Neoptera is unclear.
Probably the concept of Palaeoptera will be changed to better reflect insect evolution. In any case, three main palaeopteran lineages, traditionally treated as superorders, are recognized. Of these, the Palaeodictyopteroidea themselves might also be a paraphyletic assemblage of very basal Pterygota.
Taxonomy
Infraclass Palaeoptera
(probably paraphyletic)
Mayfly group
Ephemeroptera (mayflies) †Palaeodictyoptera group (all are extinct) †Palaeodictyoptera †Megasecoptera †Archodonata †Diaphanopterodea Dragonfly group (sometimes all are placed in the Odonata) †Protodonata or Meganisoptera †Protanisoptera †Triadophlebioptera †Protozygoptera or Archizygoptera Odonata (dragonflies and damselflies)