To the Byzantines, the lands of the distant north were self-evidently hellish. So terrible were the winters that even wolves, when they crossed the frozen seas, were reported to go blind with the cold. Unsurprisingly, then, there was nothing much for its inhabitants to do all winter except rut and procreate. In the sixth century, a bureaucrat writing in Constantinople identified Scandinavia as "a factory of peoples, a womb of nations". Three hundred years on, in 860, his warning appeared borne out, when the citizens of the great city were appalled to find a fleet of 200 Viking ships churning up the waters of the Bosphorus. Even though Constantinople itself managed to withstand the assault, the depredations inflicted on its environs were clear evidence of the wrath of God. Churches were looted, monasteries burned, captives diced into tiny pieces and dropped into the sea. Over the next 200 years, more waves of Norse pirates would follow, beating in vain against the impregnable walls of the Byzantine capital, but plundering and murdering all around them with a terrifying relish. "Fierce and merciless are these people," wailed the patriarch, "and their voice is like the roaring of the Ocean."

Yet this lament did not tell the whole story. Not every Viking who came to Constantinople was looking to loot it. Evidence for this can be found in the very heart of the city, in Hagia Sophia, the cathedral which served Byzantium as its supreme shrine. Here, in 1964, a name was discovered scratched on to its southern gallery, written in Nordic runes: "Halfdan". Then, 11 years later, a second inscription was identified: "Ari made this." Only men particularly trusted by the emperor were permitted entry to the gallery where these two Vikings, tersely but eloquently, had left memorials to their presence. What had they been doing there?

A clue to the likeliest answer can be found in a collection of sagas written at the opposite end of Europe, in Iceland. Contained within the Heimskringla, our most important source for information about Viking Norway, is the biography of its most celebrated king. Harald Hardrada, "the Hard Ruler", came to the throne late in life. As a young man, he had fled a murderous dynastic feud in his homeland for Russia, a land reliably reported to be the haunt of giants and men with mouths between their nipples, but also of booming Viking trading-posts such as Novgorod and Kiev, towns brash with frontier spirit. None of those foundations, though, could compare for sheer opportunity with Caesar's golden capital: "the Great City" or "Miklagard". Sure enough, it was in Constantinople that the exiled Hardrada finally ended up. There, he had highly prized qualities to market: his courage, his height, his proficiency with an axe. Tame a Viking, so the Byzantines had realised, and he could serve as a truly formidable mercenary. Like Halfdan and Ari, whose chance to carve their names in Hagia Sophia had evidently come while standing guard over the emperor, Hardrada had entered imperial service. Unlike Halfdan and Ari, he had fast made a name for himself that would echo as far as Iceland. He fought in Sicily, so the Heimskringla informs us, he liberated the Holy Land from the Saracens, he was fancied by an empress, he fought a dragon. By 1044, when he finally left Constantinople, he was a living legend. "Freighted with hard-won honour and gleaming gold," he found it a relatively simple matter, on his return to Norway, to make himself king.

A brooch shaped like a ship (800-1050) illustrates the Viking's taste for bling. Photograph: The National Museum of Denmark

No one before the Vikings had ever travelled so widely and to such spectacular effect. We in Britain chiefly remember Hardrada today for his role in the events of 1066, which saw him make a power-grab for the English throne, and end up dead in the battle of Stamford Bridge. Now, though, thanks to a spectacular new exhibition at the British Museum, we have been given the opportunity to place him and his fellow Norsemen in the wider context they deserve. Few will be able to visit the show and not come away with a renewed sense of stupefaction at just how far the Vikings ranged. From Iceland to north Africa, from Canada to the Caspian Sea, these were the first people in history to reach four continents. In doing so, they toppled kingdoms, discovered new worlds, and enriched as well as devastated the civilisations they preyed upon. "Bleak gales lashed prows, hard along the shoreline – but our ships, iron-shielded, rode undaunted into harbour." So sang a poet at the court of Hardrada. He understood what a Viking audience wished to hear.

It is appropriate, then, that the exhibition in the British Museum should be dominated by a ship. Roskilde 6 was discovered in the harbour area of Denmark's royal cathedral city, just west of Copenhagen, and provides an impressive centrepiece for the museum's new exhibition space. Built when Hardrada was still a boy, the ship is the longest vessel to have survived from the Viking age, and, even partially preserved, is a powerful reminder that this was a sleek and deadly weapon of war. The size of a king's dragon ship, it will not surprise Freudians to learn, was regarded as a telling measure of virility in the Viking world, and Hardrada's own was modelled on one commanded by a predecessor on the Norwegian throne, the longest and most celebrated ever built. "Its prow had a dragon's head, and the stern had a dragon's tail, and the bows were inlaid with gold." No trace of gilding survives on Roskilde 6, but there is plenty of evidence elsewhere in the exhibition of the Viking obsession with bling. Arm rings and pendants, necklaces and brooches: all of them blaze, even behind display cases, in a way that photographs can never hope to capture. This, perhaps is the most thrilling aspect of all: the way in which, by pairing longship and precious metals, it makes visual the synergy between sea power and treasure hunting that is such a perennial theme in the written record. "Far they travelled, and manfully, to win gold." So reads the Gripsholm Stone, a memorial to a Swedish adventurer who had followed the aptly nicknamed Ingvar the Far-Travelled to the shores of the Caspian Sea. "Out east they fed the eagles, then died in the south, in Serkland" – the land of the Saracens.

The longship Roskilde 6 is the longest vessel to have survived the Viking age

That Ingvar and his men were perishing on the northern borders of the Caliphate in 1042, even as Hardrada was leaving Constantinople, brings home to us that the Viking field of operations in Eurasia was almost as extensive as in the North Atlantic. For a long time, indeed, the Islamic dirham constituted the standard of wealth across Scandinavia. Arabic coins had begun to reach there even before the first raids were launched on Britain, and 84,000 silver coins from the Islamic world have been found in Sweden alone. Two hoards on display in the exhibition highlight the role played by the Vikings in what might euphemistically be termed the redistribution of global wealth. The Vale of York hoard, which was buried in the late 920s, demonstrates that silver from the Caliphate was circulating freely during the Viking age as far west as Britain. A second hoard, found in Lake Ilmen in Russia, contains, among all its various other treasures, three gilded coins from India – a potent illustration of just how far the vortex of Viking gold-hunger could reach. Finds made at a third site, the Swedish island of Helgö, include the head of an Irish crozier, a Coptic ladle and a bronze statuette of the Buddha. It mattered little to a ruler like Hardrada how he reaped his treasure, whether by exchange or plunder, just so long as he had enough of it to dispense to his followers, and to be praised by poets as a wealth-wounder, a ring-tosser, a foe of gold and silver. "By their clothing, their glittering armlets, you can tell they are the king's friends." Austerity held no appeal to a Viking. Only dragons hoarded wealth. Scandinavia, as a result of its piratical and flamboyant Keynesianism, sucked in treasure from the outermost limits of the world.

Thirty-four years ago, when the British Museum staged its last exhibition on the Vikings, that was much less evident. During the cold war, opportunities for western scholars to research the Viking penetration of Eurasia along the great river systems of the Volga and the Dnieper were in the deep freeze. Today, even though enthusiasm in Russia for acknowledging the contribution of Scandinavian colonists to early Kiev and Novgorod is not all that it might be, a dialogue is taking place between museums on either side of the vanished iron curtain. Unsurprisingly, the curators of the present exhibition capitalise on the potential of this collaboration. The Viking expansion into the virgin territory of Iceland and beyond has been downplayed, the roads to Serkland and Miklagard emphasised. Artefacts from Greenland and Newfoundland lose out to those from the east. Since many of these are being shown for the first time in Britain, it is hard to regret the curators' decision. This is a show less about the Vikings themselves than about their interaction with other cultures.

Simultaneously, though, it has a second focus: a field of research where, again, much has changed over the past 30 years. As the exhibition's subtitle, Life and Legend, suggests, the Vikings occupy an ambivalent dimension in the popular imagination, midway between history and myth. In part, this is because most of the contemporary sources for their activities derive from their victims, whose horror at "the ravages of heathen men" was hardly conducive to arriving at a measured perspective on late Iron Age Scandinavia. It also reflects something else, though: the sheer potency of those Norse traditions that did end up preserved in written form. The Heimskringla, with its stirring account of the exploits of Norway's various kings, was set down in the 13th century, long after the events it describes. How much trust, then, can we put in it as a source for what might actually have happened in the time of the Vikings? Time was when historians took its essential veracity for granted. Then scepticism set in. Had Hardrada really liberated the Holy Land, as the Heimskringla claimed, or fought a dragon? Clearly not. Yet the pendulum of opinion ended up, perhaps, swinging too far. Recent decades have witnessed a concentrated effort by scholars to negotiate the twin poles of credulity and doubt, and to arrive at a more nuanced understanding of the sagas as a source for the Viking age. Increasingly, this has meant measuring them against the material culture of the period in which they are set. It is this that makes an exhibition such as the British Museum's so revelatory. It enables us all to join in evaluating the raw material of Viking history.

Proficiency with an axe, such as this silver-inlaid axehead (900), was prized. Photograph: The National Museum of Denmark

Fans of Chris Hemsworth movies will particularly enjoy the opportunity to explore the mysterious world of the Norse gods: Thor, Loki et al. Scholars have long appreciated that the very concept of a "religion" is, in the context of pagan Scandinavia, profoundly anachronistic. Our written sources for it date from the same period as the Heimskringla, and have been similarly distorted by a process of Chinese whispers. Not only is there an unmistakable stamp of Christian influence on legends such as the Norse apocalypse Ragnarök, but their preservation in written form has veiled from us something that in the exhibition becomes clear: the degree to which beliefs in the Viking world were always highly fluid. The gods who swagger, kill and rape their way across the pages of the Icelandic sagas were certainly worshipped in pagan Scandinavia, but not as protagonists within a stable body of legends. One particular find hints at how boundaries in Norse belief existed to be crossed as well as respected. A silver statuette, discovered five years ago in Denmark, shows a seated figure dressed in the apron and necklaces of Freyja, the goddess who had taught sorcery to mankind, but flanked by the two pet ravens of Odin, the "All‑Father", king of the gods. Which of the two deities does it represent? It could be either – or both. In the Viking world, where warriors were rumoured to turn into bears and anxious abbots reported rumours of masks that gave men the heads of dogs, shape‑shifting was taken for granted.

The most profound transformation of all, and the one that would ultimately spell the true twilight of the gods, was the triumph in the Viking heartlands of Christianity. By the time that a Danish king, Harald Bluetooth (whose unifying tendencies are recognised today when mobile phones connect to computers), bragged in the mid-10th century of forcing "the Danes to be Christian", the faith was already an old one in Scandinavia. The exhibition has a replica of the huge granite runestone on which he made his boast, complete with a brightly painted image of Christ, crucified and serpent-entangled. To kings such as Bluetooth, it was not immediately obvious that his new divine patron was so very different from the old. Like Odin, the Christian god commanded a heavenly host, promised his followers rewards after death and was capable of prodigious feats of slaughter. Not surprisingly, then, kings who had received baptism were still reported to receive the odd visit from the broad-hatted All-Father – even, though, after a night spent chatting with him, they would make a point of refusing any sides of beef that the god might have left behind.

A hundred years on from Bluetooth's conversion, though, visiting rights had been refused to Odin once and for all. Hardrada, whose half-brother, Olaf Haraldsson, would end up enshrined as Norway's patron saint, had seen enough of Byzantium to know just what splendours Christian lordship had to offer an ambitious warlord. He reigned at a decisive turning point in Scandinavian history, a moment when it was still possible to combine the pleasures of the old ways with all the rich opportunities of the new. Even as the tomb of his half-brother began to swell with pilgrims drawn from across Christendom, so Hardrada himself continued to revel in the traditional activities of Norwegian warlords: waging pointless wars against his neighbours, incinerating their towns and menacing their coastlines with showy dragon ships. Nevertheless, his invasion of England was fated to be the last one by a Viking king, and even though his grandson, the magnificently nicknamed Magnus Barelegs, did lead a successful expedition against Orkney, an era was drawing to a close. A world in which all the kings of Europe were Christian was one, it turned out, with no room for Vikings – not even royal ones. The fury of the Northmen, which for three centuries had terrorised Christendom, altered the course of history, and fashioned an enduring legend, had abated at last.