This summer, on the western edges of Britain and Ireland, was a time of gentle monsters: great black fins parading sedately off the beaches, leviathans floating in warm sea as docile as Granddad on a lilo. From Cornwall to Donegal, local papers ran stories of swimmers' and kayakers' encounters with sharks "Bigger than Jaws!" "The size of a bus!" But most of the reports went on to say that the fish – which can indeed grow to 11m, a double-decker's length – were strangely blasé about the panicky, flapping humans. In fact, they didn't seem interested at all.

The basking sharks (or the cearban, the muldoan, hoe-mother, the brigdie… every Atlantic coast has its name for them) were back. They were late this year because the sea was colder than usual. They usually appear from May in the southwest, June in the Isle of Man and July in the Hebrides. But when they did turn up it was in great numbers. By August the sharks were swarming up the Scottish coast. Fishing boats and Ribs reported near-misses. On the Oban to Barra run, the Caledonian MacBrayne ferry had to keep a special lookout so the ship could avoid schools of giants cruising the seas at a sedate 3mph. The Shark Trust, which logs sightings, announced record-breaking numbers for Scotland.

Basking sharks are Britain's elephants, our biggest animals. They're also our most mysterious. They arrive in herds and then all but disappear for decades. For long periods in the 80s and 90s it was thought they had been fished nearly to extinction. (It wasn't until 1998 that hunting them was outlawed.) Behind most of the Atlantic coast's myths of water monsters and sea snakes lie basking sharks, with their weird snouts and confusing skeletal remains. The long claspers – the male sexual organs – can look like a pair of legs, and decomposing baskers fooled several 19th-century naturalists into announcing the discovery of new species.

Big gulp: the giant fish ingests an Olympic swimming pool’s-worth of water every couple of hours while feeding. Photograph: Alamy

Though we've slaughtered them and mythologised them, we still know very little about them. We don't how long basking sharks live – it may be as much as 50 years – or their gestation period. We're only just beginning to discover where they go when the summer plankton leave British shores.

You would think something so vast could not be so obscure: they are the largest fish in the North Atlantic (and the second-biggest in the world). Their two metre-wide mouths ingest an Olympic swimming-pool's-worth of water every couple of hours while feeding. It takes half a ton of plankton to fill their stomachs, and they have livers the size of cars, but brains no bigger than an apple. They will cross the Atlantic and venture a mile deep browsing for plankton, but they're not very complicated: sea anemones with fins, says one marine zoologist. But they are gentle. In Irish Gaelic there's a saying: "As tranquil as a basking shark."

They may, in fact, be the closest thing nature has to the Pathetic Sharks of Viz magazine: harmless, oversized and keener on lounging around near the beach than doing any menacing. Their name derives from the fact that they do indeed appear to sunbathe (in England they used to be called "sunfish"), as they browse along the surface in the sunlight looking for the blooms of zooplankton they eat. They are so placid you can float alongside them in a little boat for hours. I have a friend who went snorkelling with them this summer and claims to have swum inside that vast mouth, peered down the stomach and come out again. (It should be said this sort of behaviour is not good for you or the shark.)

We used to be frightened of the baskers. When I was a child on summer holidays, we went fishing for mackerel and saithe around the rocks of the little island of Tiree in the Hebrides. There was a lot that could go wrong with our little flat-bottomed wooden boat and its temperamental Seagull outboard, and much did. But the only thing we were truly scared of were the basking sharks. When the big snouts and oily black fins appeared near the boat, we started up the outboard and ran for shore. The local fisherman had taught us to be terrified of the cearban. "They'll ram your dinghy and you'll be matchwood," we'd been told. "They'll rub themselves on it, to get rid of the barnacles, like a bullock against a post."

This inaccurate lore may have been dated back to some 30 years earlier, when a bizarre tragedy happened at Carradale on the Kintyre peninsula, 40 miles to the southwest. It made the basking shark into a national enemy. A wooden sailing dinghy full of locals and holidaymakers was dismasted and capsized by a basker that appears to have either bashed it with its tail or jumped out of the water on top of it. Two adults and a six-year-old boy drowned.

A panic ensued. The following week two passenger steamers on the Clyde reported being menaced by a large shark – one captain said it had smashed some portholes. A somewhat flippant leader ran in the Times expressing shock "that the waters round this sceptered isle are shark-infested". Questions about the "menace" were asked in the House of Commons, citing bathers "chased out of the sea" and fishermen afraid to go out on the water. The basking sharks, plankton eaters, were also accused of consuming the valuable herring. In 1939 the secretary of state for Scotland promised that Fishery Board cruisers would continue to kill all the sharks they could find.

Open season: a basking shark harpooned off the Western Hebrides. Photograph: Raymond Kleboe/Getty Images

More disastrous for the innocent basking sharks was the publicity. Lured by the promise of that one-ton liver and the valuable oil in it – which was used for everything from facial moisturiser to lighthouse mechanisms – several new basking-shark fishing operations started up. The most famous was the brainchild of the naturalist and writer Gavin Maxwell, who was to become famous for books like Ring of Bright Water.

Maxwell was not as kind to the sharks as he was to his beloved otters. Driven by a romantic vision that came to him in a shelter during the London Blitz, in 1944 he bought an island off Skye and then some ex-Royal Navy gun-boats. He built a processing factory, and with a colleague from the Special Forces set about harvesting the sharks of the Minch.

His original idea was to sell every bit of the fish – from flesh to skin and fins (basking shark fins fetch more than any other in Chinese markets). Maxwell even sold some pickled shark flesh to the United Nations, then busy providing for the hungry in war-ravaged Europe. But none of this proved economically feasible: it turned out that the only practical way to use the sharks was to harpoon them, cut out the liver while they were tied to the boat, and abandon the corpse.

Then the price of liver oil dropped. At the same time, the sharks staged one of their still-unexplained disappearing acts. By 1948 Maxwell had grown sick of the business. He sold it, having apparently killed most of the golden geese and lost his shareholders £50,000. For Maxwell, though, the story ended well: he wrote a bestseller about the adventure, Harpoon at a Venture.

For centuries, the people of Atlantic coasts have spun stories about basking sharks, not least about where they go at the end of the summer. Until recently it was believed that after the summer they sunk to the depths of the seabed, to hibernate in a state of suspended animation. But now satellite communications and modern technology are beginning to shed some more light.

Marine biologist Suzanne Henderson is working with Scottish Natural Heritage and the University of Exeter on a government-funded programme to tag sharks in the Hebridean "hotspots" – shallow waters around the islands where they have always congregated. You can see the results of this in an amazing live map (wildlifetracking.org) of the movements since July of 27 individual sharks. The project has already uncloaked some of the mysteries. The sharks do venture deep, as much as 1km, but they don't go there to have a snooze – they continue moving. And, far from hibernating, two of last year's tagged sharks turned up in the winter off the west coast of Africa, near the Canaries.

On the lookout: Gavin Maxwell in 1945 – the author set up a shark fishery on the island of Soay. Photograph: Raymond Kleboe/Getty Images

Yet, not least because the battery-powered tags last a maximum of 300 days, there is much still to discover. "We don't yet know if the population we're seeing each year is the same sharks coming back again," says Dr Henderson. That means there's no real possibility of assessing the state of the basking shark population.

Mystery still also surrounds reproduction: no scientist has ever seen a basking shark mate or give birth. But what the new data may be beginning to show is that the Hebrides are a key area for feeding and mating. This will fuel the debate over whether one of the new Marine Protected Areas should cover the sites – and to keep offshore wind farms away from the islands. "The tagging technology is leaping forward year on year: I'm sure that in a few more it will do bigger and better things."

Dr Henderson has spent much of the past few summers afloat among the sharks of the Inner Isles, watching the vast fish grazing on the zooplankton – "so elegant and graceful". She's even seen one "breech" – throw itself completely out of the water near the Skerryvore lighthouse. "An amazing sight. The energy of it! Think of a salmon leaping and then multiply that, however many thousands of times. But we've no idea yet if it's to do with courtship or if the shark is trying to get rid of parasites."

A few people think the sharks should be left obscure. "Basking sharks have a right to freedom of movement and a private life without government tagging and interference!" runs a comment on one Scottish newspaper's report of the SNH/University of Exeter work.

The natural history writer Adam Nicholson was also in the Hebrides this summer, filming whales and sharks for a BBC series. On board the boat was Howard McCrindle, the last man to fish basking sharks. Nicholson has some regrets about the new research. "They are so mysterious. People don't understand their life pattern at all… Even in this age, they are still profoundly unknown. And that's good."

You can read reports of shark sightings and log your own at baskingsharkproject.blogspot.co.uk