Temnospondyli (from Greek τέμνειν, temnein 'to cut' and σπόνδυλος, spondylos 'vertebra') or temnospondyls is a diverse ancient order of small to giant tetrapods—often considered primitive amphibians—that flourished worldwide during the Carboniferous, Permian and Triassic periods, with fossils being found on every continent. A few species continued into the Jurassic and Early Cretaceous periods, but all had gone extinct by the Late Cretaceous. During about 210 million years of evolutionary history, they adapted to a wide range of habitats, including freshwater, terrestrial, and even coastal marine environments. Their life history is well understood, with fossils known from the larval stage, metamorphosis and maturity. Most temnospondyls were semiaquatic, although some were almost fully terrestrial, returning to the water only to breed. These temnospondyls were some of the first vertebrates fully adapted to life on land. Although temnospondyls are amphibians, many had characteristics such as scales and large armour-like bony plates (osteoderms) that generally distinguish them from the modern soft-bodied lissamphibians (frogs and toads, newts, salamanders and caecilians).

Temnospondyls have been known since the early 19th century, and were initially thought to be reptiles. They were described at various times as batrachians, stegocephalians and labyrinthodonts, although these names are now rarely used. Animals now grouped in Temnospondyli were spread out among several amphibian groups until the early 20th century, when they were found to belong to a distinct taxon based on the structure of their vertebrae. Temnospondyli means "cut vertebrae", as each vertebra is divided into several parts (intercentrum, paired pleurocentra, neural arch), although this occurs widely among other early tetrapods.

Experts disagree over whether temnospondyls were ancestral to modern amphibians (frogs, salamanders and caecilians), or whether the whole group died out without leaving any descendants. Different hypotheses have placed modern amphibians as the descendants of temnospondyls, as descendants of another group of early tetrapods called lepospondyls, or even as descendants of both groups (with caecilians evolving from lepospondyls and frogs and salamanders evolving from temnospondyls). There is further disagreement about a temnospondyl origin of lissamphibians related to whether the modern groups arose from only one group (dissorophoids) or from two different groups (dissorophoids and stereospondyls). The majority of studies place a group of temnospondyls called amphibamiforms as the closest relatives of modern amphibians. Similarities in teeth, skulls and hearing structures link the two groups. Whether temnospondyls are considered part of the tetrapod crown or stem thus depends on their inferred relationship to lissamphibians.

In 2000, Adam Yates and Anne Warren defined the name Temnospondyli as applying to the clade encompassing all organisms that are more closely related to Eryops than to the "microsaur" Pantylus. By this definition, if lissamphibians are temnospondyls and Pantylus is a reptiliomorph, the name Temnospondyli is synonymous with Batrachomorpha (a clade containing all organisms that are more closely related to modern amphibians than to mammals and reptiles).

Rainer Schoch in 2013 defined the name Temnospondyli as applying to "[t]he least inclusive clade containing Edops craigi and Mastodonsaurus giganteus".

Many temnospondyls are much larger than living amphibians, and superficially resemble crocodiles, which has led many taxa to be named with the suffix -suchus. The largest taxa, which were predominantly the Mesozoic stereospondyls, had skulls exceeding one meter in length, and the entire animal would have been several meters in length (for reference, the largest living amphibian, Andrias, is about 1.8 meters in body length). Others are smaller and resemble salamanders, in particularly the amphibamiform and micromelerpetid dissorophoids.

Skulls are generally parabolic to triangular in shape when viewed from above, and they were particularly flattened in semiaquatic to aquatic taxa, with dorsally facing orbits. The skull is usually covered in pits and ridges to form a honeycomb-like pattern. One of the most recent hypotheses for the function of the dermal ornamentation is that it may have supported blood vessels, which could transfer carbon dioxide to the bones to neutralize acidic build up in the blood (early semiaquatic tetrapods would have had difficulty expelling carbon dioxide from their bodies while on land, and these dermal bones may have been an early solution to the problem). However, there are many other possible hypotheses for the purpose of the ornamentation (e.g., increasing surface area for better adhesion of the skin to the skull), and the function(s) remains largely unresolved due to the absence of this feature in lissamphibians. Some temnospondyls also exhibit raised tubercles or pustules instead of pits and grooves (e.g., the dissorophoid Micropholis, plagiosaurine plagiosaurids), and the import of this disparity is also unclear. Many temnospondyls also have canal-like grooves in their skulls called sensory sulci, the presence of which is used to infer an aquatically inclined lifestyle. The sulci, which usually run around the nostrils and eye sockets, are part of a lateral line system used to detect vibrations in water in modern fish and certain modern amphibians. Many taxa, especially those inferred to have been terrestrial, have an opening at the midline near the tip of the snout called the internarial fenestra / fontanelle; this may have housed a mucous gland used in prey capture. In zatracheids, this opening is greatly enlarged for an unknown purpose.

Homologues of most of the bones of temnospondyls are also seen in other early tetrapods, aside from a few bones in the skull, such as interfrontals, internasals and interparietals, that have developed in some temnospondyl taxa. The intertemporal, a bone common in stem tetrapods, is only found in some late Paleozoic taxa like certain edopoids and dvinosaurs. Most temnospondyls have an indentation at the back of the skull called otic notches. It has typically been inferred that this structure supported a typanum for hearing, although there is substantial variation among temnospondyls in the anatomy of this notch such that it may not have served this function in all temnospondyls, and some clades like plagiosaurids and brachyopids lack notches entirely.