Duncan I was King of Scotland between c.1034 and 1040, when he was killed in battle.
His father was Crinan, Mormaer of Atholl, and lay abbot of Dunkeld. His mother, however, was Bethoc, eldest daughter of King Malcolm II. Malcom had no sons, and accordingly a claim to the Scots throne passed through Bethoc to her son. Indeed, one version of the history (which, as with much in this period, is muffled in mist as thick as Scotland can produce) states that, through the law of tanistry, Malcolm named Duncan his successor.
Duncan was made king of the "Cumbrians", a sub-nation in the Strathclyde region, and on Malcolm's death in 1034, he took the Scots throne. He was not a popular king and he became embroiled in a feud with Macbeth, who was also descended from one of Malcolm's daughters (a different one), and who had married Gruoch, grand-daughter of Kenneth III, thereby securing two lines of claim to the throne.
Duncan also attempted to expand Scotland, by campaigns south as far as Durham, and north to the Moray Firth. In the first he suffered heavy losses and was forced to retreat; in the second he faced Thorfinn of Orkney, one of the last Norse sub-kings, and also sufferd defeat. Thorfinn made common cause against Duncan with Macbeth (leading to the fictive by Dorothy Dunnett [King Hereafter] that they were one and the same, under Norse and Christian names); in 1040 they joined forces to repel Duncan from the north and slew him.
Duncan was married, to Sybilla, apparently either a daughter of, or a cousin to, Siward, Anglo-Norse Earl of Northumbria. They had two sons, Malcolm III, who killed Macbeth and married the widow of the earl of Orkney (presumably Thorfinn) and later Margaret, daughter of the English Atheling Edgar, and Donald III -- aka Donal-bane -- who both became Kings of Scotland