Introduction: Primitive Asaphida (possibly ancestral to some other asaphine groups), including families retaining the natant hypostomal condition, as well as other ptychoparioid features; protaspides resembling those of Ptychopariida (not "asaphoid"); not all families included may be monophyletic (i.e., Anomocaroidea as described here is likely a paraphyletic group).
Cephalon: preglabellar field wide; glabella typically parallel or gently tapering, with 3 or 4 pairs of furrows more or less of ptychoparioid type, palpebral lobes long, sickle-shaped; natant hypostome, median ventral suture; natant hypostome, some approaching conterminant.
Thorax: 10-13+ segments.
Pygidium: typically large, with broad, usually concave border, 2-10 axial rings.
Families: Andrarinidae, Anomocarellidae, Anomocaridae, Aphelaspididae, Parabolinoididae, Pterocephalidae (including Housiidae).
Genera: Andrarinidae: Andrarina (/Liostracus), Groenwallia, Groenwallina.

Introduction: As in typical Asaphida with the following additional defining characteristics:
Cephalon: bearing preoccipital glabellar tubercle; glabella elongate, subparallel to tapering forward, with defined occipital ring, curved, apostrophe-like pair of basal glabellar furrows isolated within glabella; hypostome conterminant (rarely impendent)
Thorax: typically 8 segments (6 to 9 in some Ceratopygidae)
Pygidium: typically rounded and without spines, but sometimes with a terminal spine or pair of spines (e.g., Thysanopyginae).
Families: Asaphidae, Ceratopygidae.
Genera: Asaphidae: Anataphrus, Araiocaris, Asaphellus (=Asaphelloides; =Asaphoon; =Hemigyraspis; =Megalaspidella; =Plesiomegalaspis), Asaphus (=Schizophorus), Atopasaphus, Aulacoparia, Aulacoparina, Australopyge, Baltiites, Banqiaoites, Basilicus (=Basiliella; =Carinobasiliella; =Dolerobasilicus/ Basilicoides; =Mekynophrys; =Parabasilicus), Bellefontia, Birmanitella, Birmanites (=Opsimasaphus), Bohemopyge (/Ptychocheilus), Borogothus, Brachyaspis, Branisaspis, Burminresia, Charabaia, Chengkouella, Dubovikites, Ectenaspis, Ekeraspis, Emanuelaspis, Emanuelina, Eoasaphus (/Anorina), Eoisotelus, Erdelia (/Maja), Estoniites, Fuyunia, Gerasaphes, Gog, Gogiura, Golasaphus, Griphasaphus, Guohongjunia, Hazarania, Heraspis, Hoekaspis, Homalopyge, Hunjiangites, Hunnebergia, Huochengia, Iduia, Isabelinia, Isotella, Isoteloides, Isotelus (=Homotelus), Isyrakella, Isyrakopeltis, Kainisiliellina, Kayseraspis, Klabavia, Kobayashia, Lachnostoma, Lamanskytes, Lapidaria, Leningradites, Liomegalaspides, Lisogorites (=Trigonoaspis; =Tangyaia), Liushuicephalus, Lonchobasilicus (=Sinomegalaspis), Lycophron, Megalaspidella, Megalaspides (=Lannacus), Megasaphus, Megatemnoura, Megistaspidella (=Spinopyge), Megistaspis (/Megalaspis; =Megistaspinus; =Rhinaspis), Merlinia, Metaptychopyge, Metayuepingia, Mioptychopyge, Mischynogorites, Nahannia, Neoasaphus (=Trematophoris; =Multiasaphus; =Postasaphus; =Subasaphus), Neopeltis, Nerudaspis, Nileoides, Ningkianites, Niobe, Niobella (=Metoptogyrus), Niobides, Niobina, Nobiliasaphus (=Pamirotchechites), Norasaphites, Norasaphus, Norinia, Notopeltis, Ogmasaphus, Ogyginus, Ogygiocarella, Ogygiocaris, Ogygitella, Ogygites, Ogygitoides, Onchometopus, Parabellefontia, Paramegalaspis (=Dolerasaphus), Paramegistaspis (/Varvaspis), Paraptychopyge, Paratamdaspis, Parayuepingia, Penchiopsis, Platyptychopyge, Plectasaphus, Plesiyuepingia, Popovkiaspis, Popovkites, Praecoparia, Presbynileus (/Paranileus), Priceaspis (=Fitzroyaspis), Proasaphus, Promegalaspides, Protopresbynileus (/Pseudonileus HINTZE, 1953), Protoptychopyge, Proxiniobe, Pseudoasaphinus, Pseudoasaphoides, Pseudoasaphus, Pseudobasilicoides, Pseudobasilicus, Pseudobasiliella, Pseudobasiloides, Pseudogriphasaphus, Pseudogygites, Pseudomegalaspis, Pseudoptychopyge, Pseudoptyocephalus, Psilocephalina, Psilocephalops, Ptychopyge, Ptyocephalus (=Kirkella), Rhinoferus (=Lawiaspis; =Ropschiaspis), Sanbernardaspis, Shergoldina, Stegnopsis, Stenorhachis, Suriaspis, Tchukeraspis, Thysanopyge (=Basilicoides), Trigonocerca, Trigonocercella, Tsaidamaspis, Valdaites, Vogdesia, Volchovites, Xenasaphus, Xenostegium, Xinanocephalus, , Yuepingioides, Zhenganites (=Eosoptychopyge), Zoraspis, Zuninaspis.

Cephalon: Opisthoparian or marginal facial sutures, generally eyeless; glabella typically convex and pyriform, with 3 or fewer pairs of furrows, preoccipital glabellar tubercle sometimes present; usually long genal spines.
Thorax: usually 5 – 8 segments, but only 2-3 segments in progenetic Raphiophoridae, and up to 30 in Seleneceme (Alsataspididae), with long, narrow adaxial pleurae.
Pygidium: wide, typically triangular, narrow axis extending to posterior margin, border strongly declined, doublure very narrow.
Other: Asaphoid protaspis shows common ancestry; Raphiophorus is the only Trinucleioid (indeed the only representative of the order Asaphida) that continues beyond the Ordovician-Silurian boundary.
Families: Alsataspididae (including Orometopidae), Dionididae, Liostracinidae, Raphiophoridae, Trinucleidae, 
Genera: Alsataspididae: Ajrikina, Alataupleura, Araiopleura, Calycinoidia, Caputrotundum, Clavatellus, Falanaspis, Hapalopleura, Huamiaocephalus, Jegorovaia (=Hermosella), Jiangxiaspis, Orometopus, Pagometopus, Palquiella, Paracalymenemene (/Paracalymene LIU), Plesioparabolina, Pyrimetopus, Rhadinopleura, Seleneceme (=Alsataspis), Sibiriopleura, Skljarella (=Proaraiopleura), Spirantyx, Trigocephalus, Yumenaspis, Zacompsus.

Cephalon: with opisthoparian sutures, glabella typically truncate anteriorly and squat, 1-4 pairs of lateral furrows, 1p may be transglabellar; preglabellar field variable, sometimes absent, palpebral ridge typically well-defined, but separate from axial furrow (compare to sister group Remopleuroidea); genal spines typically present, of various length; median ventral suture rarely lost to secondary fusion; hypostome conterminant
Thorax: 8-12 segments, axis convex, pleurae typically wider than length of axis, typically with short, pointed ends (longer in Loganellidae).
Pygidium: micropygous to isopygous, variable shape, axis often extends majority of length, sometimes with post-axial ridge, posterior margin smooth or spined (1-5 pairs of marginal spines)
Families: Dikelocephalidae, Eurekiidae, Ptychaspididae, Saukiidae, Loganellidae.
Genera: Dikelocephalidae: Berkeia, Blandicephalus, Briscoia, Camaraspoides, Dikelocephalus, Elkia, Goumenzia, Hoytaspis, Iranella, Kasachstanaspis, Monocheilus, Olimus, Osceolia, Parabriscoia, Patalolaspis, Princetonella (/Calyptomma), Pterocephalops RASETTI, Randicephalus, Stigmacephalus, Walcottaspis.

Cephalon: with glabella expanding forward to anterior margin, effaced in later cyclopygids, may be fused with occipital ring; fixigenae reduced (except in primitive Taihungshaniidae), palpebral lobes lack distinct rims, and contact axial furrows at anterior ends, librigenae fused or separated by anterior median suture; hypostome relatively transverse, impendent, often with tripartite posterior margin; eye various sized (may be hypertrophied and convex (Cyclopyge), typically closely adjoined to glabella.
Thorax: 5-8(9?) segments.
Pygidium: medium to large (subisopygous in Nileidae), axis usually with 2-5 rings (but up to 20+ in advanced Taihungshaniidae), may be smooth, or with indistinct furrows.
Families: Cyclopygidae, Nileidae, Taishunghaniidae
Genera: Cyclopygidae: Amicus, Aspidaeglina, Circulocrania, Cyclopyge (/Egle/Aeglina), Degamella, Ellipsotaphrus, Emmrichops, Gastropolus (=Lisogoraspis), Girvanopyge (=Cremastoglottos; =Gamops; =Nanlingia), Heterocyclopyge (=Selenoptychus), Microparia (=Gallagnostoides), Novakella (=Incisopyge), Paramicroparia, Phylacops, Pricyclopyge (=Bicyclopyge), Prospectatrix, Psilacella, Quadratapyge, Sagavia, Symphysops, Waldminia, Xenocyclopyge.

Cephalon: with opisthoparian sutures, glabella bulges tranversely anterior of occipital ring, with up to 3 pairs lateral furrows, eyes medium to very large, with narrow, wire-like socle, palpebral rims inflated, deep rim furrows, extending into axial furrows anteriorly; genal spines present.
Thorax: 9-12 segments, axis convex, pleural furrows diagonal, pleural tips typically point backward.
Pygidium: with spinose margin, spines flattened, united at bases, extending to axis; convex axis not extending to posterior margin, pleural field flat, typically furrowed and backward curving. Surface variously sculptured or granulose.
Families: Auritamiidae, Bohemillidae, Hungaiidae (including Dikelokephalinidae), Idahoiidae (including Loganellidae), Remopleurididae (including Kainellidae).
Genera: Auritamidae: Auritama, Metopotropis.