I wrote a post about vikings here a while back. I wrote quite a bit about their accomplishments:

During the first decades of the Viking Age, the vikings were primarily raiders. The aforementioned raid on Lindisfarne was only the first of many raids in the British Isles; the villages and monasteries along the coasts and rivers of Britain, Ireland and Francia (present day France and Germany) were prime targets for raiders seeking wealth. When they weren't raiding, many vikings were farmers back home in Scandinavia. This all changed in AD 865, when the Great Heathen Army, allegedly led by the sons of Ragnar Loðbrok (meaning Hairy-Breeches), a legendary figure who, as of late, has become known to the public as the protagonist of the fantasy TV show Vikings, invaded England.

Unlike the raiders that had plagued Britain for 70 years, the Great Heathen Army, composed of Danes hungry for land, intended to conquer England for themselves. During this time, England was divided into four kingdoms; Northumbria, Mercia, East Anglia and Wessex. Over the following decade, they went on to conquer the English kingdoms. Only the emergence of King Alfred of Wessex, later Alfred the Great, in the 870s, put an end to the Great Heathen Army.

After a series of military victories, ending with the Battle of Edington in 878, Alfred signed a treaty with the vikings, now led by a man named Guthrum, declared that the Danes would withdraw from Wessex immediately and that Guthrum would be baptised. Some years later, a second treaty between Alfred and Guthrum further served to establish the Danelaw, the parts of present day England which would be ruled by the Danes. A map of the Danelaw can be found here. Alfred's children and grandchildren would later on, in the early-to-mid 900s, retake the lands under the Danelaw.

In 1015, King Cnut (or Canute) the Great of Denmark (and later Norway) invaded England, and by 1016 he had become king. His two sons would go on to inherit England, but both died within a decade of their father, leaving no heirs, and the crown instead returned to Edward the Confessor, a descendant of Alfred the Great.

In addition to England, both Danish and Norse vikings went on to settle parts of Ireland; Dublin was founded by viking settlers. There was also a considerable amount of Norse settlers in the northern parts of the British Isles, as shown in the 2001 TV documentary series Blood of the Vikings, where they found that Norwegian genetic heritage was highly common in the Shetlands and Orkneys, as well as Cumbria in northwestern England.

Norse vikings also settled Iceland in the late 9th century, and a century later, a Norse viking named Erik the Red colonised Greenland, and from there they went on to discover North America (with some expeditions led by Erik the Red's son, Leif Eriksson), but ultimately no successful attempts at settling North America were made.

Another notable viking settler is Rollo (Possibly Hrolfr in Old Norse). His background is disputed, but it is thought that he was either a Norseman or Dane who was exiled from his home after making an enemy of the king. What is known is that he frequently raided present day France during the late 800s and early 900s, besieging Paris multiple times. After an unsuccessful siege of Chartres in 911, he pledged allegiance to the King of France. In return for swearing to protect France against other vikings, the king made him ruler of Normandy.

Finally, we come to the Swedish vikings. Unlike their Norse and Danish kin, they were not quite as predisposed to raiding and conquering; instead they became merchants, trading with the people on the other side of the Baltic Sea (this is not to say that there were no Swedish raiders; we even know that there were Swedes who went west, to England, mainly based on runestones). According to Russian sources, the people of Novgorod invited a Swedish tribe known as the Rus (the East Slavic Primary Chronicle claims that the Rus were a separate people from the Swedes, but I can find no other sources supporting this claim) to come and rule them. Rurik, the leader of these vikings, founded the Rurik dynasty which, after conquering Kiev, became the rulers of Kievan Rus' and later the Grand Duchy of Moscow and ultimately founded the Tsardom of Russia.

In the east, the vikings were known as varangians or varyags. They even went as far east as the Byzantine Empire, where they would form the Varangian Guard, bodyguards to the Byzantine emperor. One member (and commander) of the Varangian Guard was Harald Hardrada, who had been exiled by Cnut the Great, having fought against Cnut alongside his half-brother, who had been the King of Norway before Cnut had taken the throne. When he returned to Norway, some years after the deaths of Cnut and his sons, he had amassed a considerable amount of wealth and managed to take the throne for himself.

Harald Hardrada spent some twenty years as King of Norway, laying claim to both the Danish and the English crowns in the process. Despite successful campaigns, he was unsuccessful in conquering Denmark. In 1066 he set out for England, landing near York. He defeated the English at the Battle of Fulford, but died five days later at the Battle of Stamford Bridge, and with him the Viking Age ended. Still, the legacy of the vikings would live on; barely three weeks after Hardrada's death at Stamford Bridge, Duke William of Normandy, a direct descendant of Rollo, defeated the English at the Battle of Hastings, to some degree fulfilling the dream of a viking-ruled England.