Donnchad mac Crínáin (Modern Gaelic: Donnchadh mac Crìonain) anglicised as Duncan I, and nicknamed An t-Ilgarach, "the Diseased" or "the Sick (died 14 August, 1040) was king of Scotland (Alba). He was son of Crínán, hereditary lay abbot of Dunkeld, and Bethóc, daughter of king Malcolm II of Scotland (Máel Coluim mac Cináeda).
Unlike the "King Duncan" of Shakespeare's Macbeth, the historical Duncan appears to have been a young man. He followed his grandfather Malcolm as king after the latter's death on 25 November, 1034, without apparent opposition. He may have been Malcolm's acknowledged successor or tánaise as the succession appears to have been uneventful. Earlier histories, following John of Fordun, supposed that Duncan had been king of Strathclyde in his grandfather's lifetime, ruling the former Kingdom of Strathclyde as an appanage. Modern historians discount this idea.
Another claim by Fordun, that Duncan married the sister, daughter or cousin of Sigurd the Dane, Earl of Northumbria, appears to be equally unreliable. An earlier source, a variant of the Chronicle of the Kings of Alba (CK-I), gives Duncan's wife the Gaelic name Suthen. Whatever his wife's name may have been, Duncan had at least two sons. The eldest, Malcolm III (Máel Coluim mac Donnchada) was king from 1057 to 1093, the second Donald III (Domnall Bán, or "Donalbane") was king afterwards. Máel Muire, Earl of Atholl is a possible third son of Duncan, although this is uncertain.
The early period of Duncan's reign was apparently uneventful, perhaps a consequence of his youth. Macbeth (Mac Bethad mac Findláich) is recorded as his dux, literally duke, but in the context — "dukes of Francia" had half a century before replaced the Carolingian kings of the Franks and in England the over-mighty Godwin of Wessex was called a dux — this suggests that Macbeth was the power behind the throne.
In 1039, Duncan led a large Scots army south to besiege Durham, but the expedition ended in disaster. Duncan survived, but the following year he led an army north into Moray, traditionally seen as Macbeth's domain. There he was killed, at Pitgaveny near Elgin, by his own men led by Macbeth, probably on 14 August, 1040.