I say "probably" not the title not because there's any question about whether hagfish are disgusting–they are, they really are–but because there's been some debate in the past about whether hagfish are vertebrates. Hagfish, as you may already know, are superficially eel-like marine animals that, together with the lampreys, are one of the two living lineages of 'jawless fish'. Their skeleton is both completely cartilaginous decidedly rudimentary: they even lack a developed spine, instead retaining the fluid-filled notochord throughout their life. They do possess a brain-case, as well as some appendicular cartilages that provide support for the fins. Around the mouth are a set of muscularly-controlled tooth-plates together with short sensory tentacles. Hagfish have no eyes; instead, they find their way about primarily through the use of a single large nostril in the middle of the head. Along the underside of the body run a series of glands capable of producing a truly mind-bending amount of mucus. As noted by Martini & Flescher (2002), "A single live individual hagfish can turn a 2 gallon pail of water into a gelatinous mass within a few minutes". Most hagfish seem to be in the one or two feet range size-wise, but the New Zealand species Eptatretus goliath was described from a single monster specimen a bit over 1.25 metres long (Mincarone & Stewart 2006). In contrast, the hydrothermal vent inhabitant Eptatretus strickrotti is only just over a foot long and built like a swimming shoelace (Møller & Jones 2007).
Martini & Flescher (2002) summarised the lifestyle of the Atlantic hagfish Myxine glutinosa (or probably the western Atlantic hagfish M. limosa which they regarded as synonymous with the eastern Atlantic M. glutinosa), which I'm guessing is fairly typical of the group. Atlantic hagfish spend most of their lives buried in burrows in muddy sea-bottoms (the technical term for the type of sediment they prefer is 'flocculent', which is a wonderful word to say), emerging primarily to feed. A large part of their diet is obtained by predating small animals such as crustaceans. They are most notorious, though, as scavengers. Hagfish will emerge in large numbers to feed on any animal corpses that sink within their range. Though they are capable of tearing off external chunks of flesh (more on that in a moment), they are not able to do so efficiently so they prefer to focus on the softer internal organs when they can. This they do by worming their way into the carcasse through a convenient orifice such as the mouth or anus and enjoying the laid-on buffet within. The reproduction of hagfish is poorly known. The Royal Academy of Copenhagen offered an award in 1864 to the first person to describe the details of hagfish nooky; the offer was withdrawn in the 1980s, still unclaimed. Female hagfish have been caught with developing eggs, up to 30 at a time, connected in a string by velcro-like hooks. The absence of any sort of obvious intromittent organ in the male suggests that fertilisation is external, but anything beyond that is a mystery.
Their lack of a rigid skeleton makes hagfish capable of some behaviours that would be beyond other vertebrates. One of these is referred to as 'knotting' and it is exactly what it sounds like. The hagfish makes a loop with its body through with it sticks its tail, quite literally tying itself in a knot. By pulling itself through itself, it can move the knot up the body until the head pops out at the other end. One reason it may do this is to clean itself; for instance, a hagfish may drown in its own mucus if not given the opportunity to remove it (so that single live individual in the two-gallon bucket is probably not live any more). Another reason is that the knot can be used to push against something, such as when the hagfish wants to escape from an enclosed space. When feeding on something large and solid (such as the aforementioned external scavenging), the hagfish will latch on with its tooth-plates and then form a knot to push against it until eventually it tears away with a mouthful of food.
Hagfish can be abundant in some areas, make them an important part of the local ecosystem. They may be regarded as a nuisance in fisheries, attacking fish caught on lines and traps and reducing their commercial value. However, hagfish are also caught for food in some parts of the world (particularly in east Asia) and their skins are cured to produce a soft textile known somewhat euphemistically as 'eelskin'.
About sixty species of hagfish are currently recognised around the world, usually classified in a single family Myxinidae. Most are divided between two subfamilies (sometimes recognised as separate families), the Myxininae and Eptatretinae. Myxininae have a single external gill opening whereas Eptatretinae have multiple gill openings. A phylogenetic analysis of the hagfish by Fernhom et al. (2013) found a couple of species previously assigned to Eptatretus to probably sit outside the Myxininae-Eptatretinae clade and transferred them to a new genus Rubicundus in its own small subfamily, differing from other hagfish in having the single nostril on a short tubular snout.
As alluded to above, there has been some debate about the affinities of hagfish. Though superficially similar to the other living group of 'jawless fishes', the lampreys (largely through both being eel-like in form), hagfish are very different in the anatomical details, and at the very least the two lineages have been separate for a very long time. Because of their lack of a number of derived features, hagfish were suggested to be the sister lineage of all other vertebrates, leading to the observation that it was not really appropriate to classify a lineage that did not have and probably never had vertebrae as 'vertebrates'. As such, hagfish became regarded as the closest relatives of vertebrates rather than vertebrates themselves. However, molecular studies of vertebrate phylogeny have pretty much universally identified hagfish as forming a clade with lampreys after all, implying that the 'primitive' features of hagfish probably represent secondary losses. When constrained as a clade in morphological analyses, nevertheless, the hagfish-lamprey group remains basal in vertebrates: most if not all of the fossil groups of 'jawless fish', particularly those with an outer covering of bony plates, are more closely related to the jawed fishes than to hagfish or lampreys.
Not surprisingly for something without much of a skeleton, the fossil record of hagfish is pretty minimal. A species from the Carboniferous Mazon Creek lagerstätte, Myxinikela siroka, is likely to be a stem-hagfish; a couple of other fossils from the same formation have also been suggested as candidates. Myxinikela was broadly similar to a modern hagfish, the most obvious difference being that it was shorter and more cigar- or banana-shaped than eel-like (I can't really imagine it being able to tie itself in knots). Some authors have also suggested similarities between the braincase of hagfish and that of Palaeospondylus, an unusual eel-like vertebrate from the Middle Devonian of Scotland whose confusing assortment of features has lead to it being seen at one time or another as a jawless fish, a degenerate bony fish that failed to develop bone, or even a larval amphibian (Janvier 2015). The most obvious difference between Palaeospondylus and a hagfish is that Palaeospondylus possessed a complete cartilaginous skeleton, but the molecular phylogenies suggest that may not be the problem it would have previously been assumed to be...
Fernholm, B., M. Norén, S. O. Kullander, A. M. Quattrini, V. Zintzen, C. D. Roberts, H.-K. Mok & C.-H. Kuo. 2013. Hagfish phylogeny and taxonomy, with description of the new genus Rubicundus (Craniata, Myxinidae). Journal of Zoological Systematics and Evolutionary Research 51 (4): 296–307.
Janvier, P. 2015. Facts and fancies about early fossil chordates and vertebrates. Nature 520: 483–489.
Martini, F. H., & D. Flescher. 2002. Hagfishes. Family Myxinidae. In: Collette, B. B., & G. Klein-MacPhee (eds) Bigelow and Schroeder's Fishes of the Gulf of Maine 3rd ed. pp. 9–16. Smithsonian Institution Press: London.
Mincarone, M. M., & A. L. Stewart. 2006. A new species of giant seven-gilled hagfish (Myxinidae: Eptatretus) from New Zealand. Copeia 2006 (2): 225–229.
Møller, P. R., & W. J. Jones. 2007. Eptatretus strickrotti n. sp. (Myxinidae): first hagfish captured from a hydrothermal vent. Biol. Bull. 212: 55–66.