Uí Ímair in the context of "Kingdom of Dublin"

The Norse–Gaels (Old Irish: Gall-Goídil; Irish: Gall-Ghaeil; Scottish Gaelic: Gall-Gàidheal; Manx: Goal-Gael, 'foreigner-Gaels') were a people of mixed Gaelic and Norse ancestry and culture. They emerged in the Viking Age, when Vikings who settled in Ireland and in Scotland became Gaelicised and intermarried with Gaels. The Norse–Gaels dominated much of the Irish Sea and Scottish Sea regions from the 9th to 12th centuries. They founded the Kingdom of the Isles (which included the Hebrides and the Isle of Man), the Kingdom of Dublin, the Lordship of Galloway (which is named after them), and briefly ruled the Kingdom of York (939–944 AD). The most powerful Norse–Gaelic dynasty were the Uí Ímair or Ivar dynasty.

Over time, the Norse–Gaels became ever more Gaelicised and disappeared as a distinct group. However, they left a lasting influence, especially in the Isle of Man and Outer Hebrides, where most placenames are of Norse–Gaelic origin. Several Scottish clans have Norse–Gaelic roots, such as Clan MacDonald, Clan MacDougall and Clan MacLeod. The elite mercenary warriors known as the gallowglass (gallóglaigh) emerged from these Norse–Gaelic clans and became an important part of Irish warfare. The Viking longship also influenced the Gaelic birlinn and longa fada, which were used extensively until the 17th century. Norse–Gaelic surnames survive today and include MacIvor, MacAskill, and [Mac]Cotter.

The Kingdom of the Isles, also known as Sodor, was a Norse–Gaelic kingdom comprising the Isle of Man, the Hebrides and the islands of the Clyde from the 9th to the 13th centuries. The islands were known in Old Norse as the Suðreyjar, or "Southern Isles" as distinct from the Norðreyjar or Northern Isles of Orkney and Shetland. In Scottish Gaelic, the kingdom is known as Rìoghachd nan Eilean. The territory is sometimes called the Kingdom of Mann and the Isles, although only some of the later rulers claimed that title. The historical record is incomplete, and the kingdom was not a continuous entity throughout the entire period. At times the rulers were independent of external control, although for much of the period they had overlords in Norway, Ireland, England, Scotland or Orkney. At times there also appear to have been competing claims for all or parts of the territory. The islands have a total land area of over 8,300 square kilometres (3,205 sq mi) and extend for more than 500 kilometres (310 mi) from north to south.

Viking influence in the area began in the late 8th century, and whilst there is no doubt that the Uí Ímair dynasty played a prominent role in this early period, the records for the dates and details of the rulers are speculative until the mid-10th century. Hostility between the Kings of the Isles and the rulers of Ireland, and intervention by the crown of Norway (either directly or through their vassal the Earl of Orkney) were recurring themes.

Olaf Guthfrithson or Anlaf Guthfrithson (Old Norse: Óláfr Guðrøðsson [ˈoːˌlɑːvz̠ ˈɡuðˌrøðsˌson]; Old English: Ánláf; Old Irish: Amlaíb mac Gofraid; died 941) was a Hiberno-Scandinavian (Irish-Viking) leader who ruled Dublin and Viking Northumbria in the 10th century. He was the son of Gofraid ua Ímair and great-grandson of Ímar, making him one of the Uí Ímair. Olaf succeeded his father as King of Dublin in 934 and succeeded in establishing dominance over the Vikings of Limerick when he captured their king, Amlaíb Cenncairech, in 937. That same year he allied with Constantine II of Scotland in an attempt to reclaim the Kingdom of Northumbria which his father had ruled briefly in 927. The forces of Olaf and Constantine were defeated by the English led by Æthelstan at the Battle of Brunanburh in 937.

Olaf returned to Ireland in 938 but after Æthelstan's death the following year Olaf left for York where he was quickly able to establish himself as king, with his brother Blácaire mac Gofraid being left to rule in Dublin. Olaf and Æthelstan's successor Edmund met in 939 at Leicester where they came to an agreement regarding the division of England between them. This agreement proved short-lived, however, and within a few years Vikings had occupied the Five Boroughs of Derby, Leicester, Lincoln, Nottingham and Stamford. Olaf died in 941 and was succeeded in Northumbria by his cousin Olaf Cuaran. At the time of his death, the Irish annals title him "king of Danes" and "king of the Fair Foreigners and the Dark Foreigners".