Introduction: A large, heterogenous order with primitive origins and problematic classification, with specialized offshoots that are hard to frame within a general diagnosis. 
Cephalon: typically with opisthoparian facial sutures, with gently forward-tapering simple glabella bearing a broad, rounded front, usually with 3 pairs of rather narrow parallel glabellar furrows; natant hypostome.
Thorax: typically large with 8+ thoracic segments.
Pygidium: quite variable, but typically with a small pygidium bearing a border (Cambrian) or a larger pygidium with or without border (post-Cambrian). 
Occurrence: Lower Cambrian to Upper Ordovician.
Suborders: Ptychopariina, Olenina, (formerly also included Harpina, which is now treated as a full Order Harpetida).

Introduction: Primitive Ptychopariida, a large and extremely varied group. 
Cephalon: glabella usually tapering with 3 pairs of glabellar furrows, sutures typically opisthoparian (but some proparian, and blind forms marginal); anterior sutures usually convergent to slightly divergent, posterior sections moderately to highly divergent; eyes usually present, medial, and near glabella; usually blade-like genal spines present. 
Thorax: generally long, relative to pygidium. 
Pygidium: variable, but typically smaller than thorax.
Superfamilies: Ellipsocephaloidea, Ptychoparioidea (see below)
Families: (listing those for which suborder assignment is uncertain) Ityophoridae, Catillicephalidae, Raymondinidae, Avoninidae, Plethopeltidae.

Superfamily Ellipsocephaloidea
Cephalon: glabella tapering forward, or subparallel or slightly expanding forward; up to 5 pairs of lateral furrows, eye ridges present.
Thorax: generally 12-16 thoracic segments.
Pygidium: small, unremarkable.
Other: includes some of the oldest trilobites in the fossil record (members of the Bigotinidae).
Families and Genera: Agraulidae: Agasella, Agaso, Agraulos (/Arion; /Arionius; =/Arionides /Arionellus; =Agrauloides), Batenoides, Chittidilla (=Diandongaspidella /Diandongaspis), Chondroparia, Clemenella, Conagraulos, Elankaspis, Lenagraulos, Litavkaspis, Metagraulos, Micragraulos, Mungyongia, Parachittidilla (=Amurticephalus), Paragraulos, Paraplesiagraulos, Phymaspis, Plesiagraulos, Poriagraulos, Proampyx, Protochittidilla, Pseudoeteraspis, Pseudoplesiagraulos, Qiannanagraulos, Shahaspis, Skreiaspis, Stembergaspis, Taiganella, Tetragonocephalus, Tholus, Tianjingshania, Veragraulos, Wutaishania.

Superfamily Ptychoparioidea
Cephalon: typically with well-defined border, glabella tapering forward, preglabellar field present, opisthoparian sutures, natant hypostome, and genal spines; but exceptions include eyeless forms (Conocoryphidae), proparian forms (Norwoodiidae), rounded genal angles and gonatoparian sutures (Menomoniidae).
Thorax: typically 12-17 segments.
Pygidium: typically micropygous, transverse, pleural field nearly flat, with distinct pleural grooves; but exceptionally isopygous (e.g., some Asaphiscidae, Cooselidae).
Families and Genera: Acrocephalitidae HUPE, 1953
Acrocephalella, Acrocephalinella, Acrocephalites (=Acantholenus), Acrodirotes, Afghancephalites, Asturiaspis, Brutaspis, Cermataspis, Decus, Diceratocephalina, Elatilimbus, Ijacephalus, Kepisis, Mansiella, Pseudacrocephalaspina, Siligerites, Toxotina, Trifonella.

Cephalon: with narrow border, opisthoparian sutures (exceptionally proparian), genal angle rounded or spined, glabella tapering forward with simple, sigmoid, or bifurcate furrows (furrows absent in some forms); librigenae yoked (fused by a narrow strip of cephalic doublure), or rarely, separated by anterior median suture; eyes small to medium, usually with distinct eye ridges. [Members of Olenina form the foundation for the Order Olenida (Adrain 2011) and will be extracted from Ptychopariida at this website in the near future.]
Thorax: 9-24 segments.
Pygidium: micropygous to subisopygous, with or without marginal spines.
Other: typically with extremely thin cuticle.
Family: Olenidae (subfamilies Leptoplastinae, Oleninae, Triarthrinae, Pelturinae).
Genera: Olenidae: Acerocare, Acerocarina (/Cyclognathus), Aciculolenus, Anaximander, Angelina (=Keidelaspis), Apoplanias, Asilluchus, Baikonuraspis, Balnibarbi, Bienvillia (=Diatemnus; =Mendoparabolina), Boeckaspis (/Boeckia BROGGER; =Sphaerophthalmella), Bondarevites, Bulbolenus, Chekiangaspis, Cloacaspis, Ctenopyge, Cyclognathina, Danarcus, Desmetia, Eoctenopyge, Euonchonotina, Eurycare, Granitzia, Hancrania, Helieranella, Highgatella, Huangshiaspis, Hunanolenus, Hypermecaspis (=Spitsbergaspis), Inkouia (=Agalatus), Isidrella, Jujuyaspis (=Alimbetaspis), Leiobienvillia, Leptoplastides (=Andesaspis; =Beltella; =Chunkingaspis; =Parabolinopsis; =Rampartaspis), Leptoplastus, Leurostega, Magnomma, Mesoctenopyge, Moxomia, Neoolenus, Neoparabolina, Nericiaspis, Olenus (=Simulolenus), Orkekeia, Parabolina (/Odontopyge), Parabolinella, Parabolinina, Parabolinites, Paraolenus, Paraplicatolina, Peltocare, Peltura (/Anthes; =Anopocare), Pelturina, Plicatolina, Plicatolinella, Porterfieldia, Prohedinella, Protopeltura, Psilocara, Remizites, Rhodonaspis, Saltaspis, Shihuigouia, Sphaerophthalmus, Svalbardites, Talbotinella, Triarthrus (/Brongniartia EATON), ?Ullaspis, Westergaardia (=Sphaerophthalmoides), Westergaardites, Wujiajiania.

Uncertain Suborder or Polyphyletic Families:

Avoninidae: Avonina, Wongia.

The genera listed here are from Jell & Adrain 2003.