Well I’ll be: a new paper in Science (click on reference below; free access with legal Unpaywall app, pdf is here) suggests that the Greenland shark, Somniosus microcephalus, is the longest-living vertebrate known to science. While there are big error bars around age estimates, it’s likely that many live at least 300 years, and some may live up to 500 years! Moreover, females don’t appear to reproduce until they’re about 150 years old. I’m stupefied!
But first a bit about this animal. Not a lot is known about it save for its geographic range, is in the northern waters of the Northern Hemisphere. Here’s the range given by Wikipedia:
And Wikipedia says this about its diet:
The Greenland shark is an apex predator and mostly eats fish. It has never been observed hunting. Recorded fish prey have included smaller sharks, skates, eels, herring, capelin, Arctic char, cod, rosefish, sculpins, lumpfish, wolffish, and flounder.
Greenland sharks have also been found with remains of seals, polar bears, horses, moose, and reindeer (in one case an entire reindeer body) in their stomachs. The Greenland shark is known to be a scavenger, and is attracted by the smell of rotting meat in the water. The sharks have frequently been observed gathering around fishing boats. It also scavenges on seals.
Although such a large shark could easily consume a human swimmer, the frigid waters it typically inhabits make the likelihood of attacks on humans very low, and no cases of predation on people have been verified.
As for its behavior, not much is known, as it’s rarely observed. It hasn’t actually been seen hunting, and nobody knows when and where the offspring are born.
Here’s one of the behemoths: they’re about 4-5 meters long and grow only about 1 cm per year, so that already implies that they’re old.
How did they figure out how old these things were? (The estimates are based on 28 female specimens ranging from 0.8 to 5.0 meters long.) The age determination was done in a clever way: using proteins in the eye nucleus of adults. These proteins are laid down in the embryos, and they don’t turn over during the shark’s lifetime. Thus the level of carbon-14 (radioactive carbon) in the nuclei reflect the amount of carbon-14 present in the water when the shark was an embryo. (The carbon in those proteins comes from the fish eaten by mom, and the carbon in the fish ultimately comes from the carbon in the water.)
This fact was combined with the fact that a pulse of radioactive carbon was injected into the atmosphere (and oceans) by nuclear bomb testing in the late 1950s and early 1960s. Although carbon-14 decays radioactively, its half-life is about 5700 years, so its levels in the ocean remain high compared to before the bomb testing. Here’s a graph from the paper showing the C-14 spike in the early 1960s in inorganic (carbonates) and organic (dietary) bits of northern Atlantic fish and sharks:

If a Greenland shark’s eye nuclei showed the high levels of “spike” C-14 attained by the species above, then, it must have been a fetus in the early 1960’s; that is, it must be less than about 60 years old. Without the spike, it must be older than 60 years.
But how much older? It turns out that there’s a “Marine13” age curve in which levels of carbon-14, which fluctuate naturally over time, can be correlated with past times by using data from tree rings, corals, plant “macrofossils”, and other carbon-containing materials of known age. That data was used to determine the ages of sharks of various sizes.
The age estimates were themselves given error estimates using Bayesian methods, with these limits incorporating estimates about growth rates, size at birth, and age estimates from one shark (#3 in the graph below) that had an intermediate level of C-14 between past background levels and bomb-testing “spike” levels. Old #3 was thus considered to be about 50 years old when it was sampled in 2012.
The graph below shows the estimated ages of the 28 sharks with 95.4% Bayesian confidence intervals (shown in dark blue) plotted against the length of each shark, so there are 28 graphs. As expected, the bigger sharks are older. (Ages are given by estimated dates of birth, which go back several hundred years).

The upshot is that these things can get old: the age estimates (with error limits) of the two biggest sharks were 335 ± 75 years and 393 ± 120 years, which suggests that some sharks could be as much as 500 years old. But we can be confident that the big ones are about 300 years old.
Further, because we know that females don’t reproduce until they’re about 4 meters long (this is presumably because no sharks smaller than that have had fetuses),the age of first reproduction would be about 156 ± 22 years.
These estimates show that the Greenland shark is the longest-lived vertebrate known to science. What about the runners-up? Well, the paper gives that information at the end (the quahog is of course a mollusc, but it appears to be the longest-living animal):
Our results demonstrate that the Greenland shark is among the longest-lived vertebrate species, surpassing even the bowhead whale (Balaena mysticetus, estimated longevity of 211 years). The life expectancy of the Greenland shark is exceeded only by that of the ocean quahog (Arctica islandica, 507 years). Our estimates strongly suggest a precautionary approach to the conservation of the Greenland shark, because they are common bycatch in arctic and subarctic groundfish fisheries and have been subjected to several recent commercial exploitation initiatives.
If sharks really can live 500 years, they would have been youngsters when Erasmus was writing and plagues of dancing mania broke out in Europe.

