Marriages and issue
His first wife was Elizabeth Mure, by her he had at least ten children:

King Robert III of Scotland (d. 1406), originally called John
Robert Stewart, 1st Duke of Albany (1339–1420)
Alexander Stewart, 1st Earl of Buchan, and Lord of Badenoch, "the Wolf of Badenoch" (1343–1405)
Margaret Stewart, married John of Islay, Lord of the Isles
Walter Stewart (d. 1362), married Isabella, Countess of Fife
Marjory Stewart, married first John Dunbar, 5th Earl of Moray, second Alexander Keith
Johanna Stewart, married in 1373 Sir John Keith, in 1379 Sir John Lyon, in 1384 Sir James Sandilands
Isabella Stewart, married first James Douglas, 2nd Earl of Douglas, married second David Edmondstone
Katherine Stewart, married Sir Robert Logan of Grugar, later also of Restalrig, Lord High Admiral of Scotland
Elizabeth Stewart, married Sir Thomas Hay, Lord High Constable of Scotland
By his second wife, Euphemia de Ross, daughter of Aodh, Earl of Ross, and widow of the 3rd Earl of Moray, formerly his colleague as regent, he had five children:

David Stewart, 1st Earl of Caithness (d. bef. 1389)
Walter Stewart, 1st Earl of Atholl (d. 1437)
Margaret Stewart
Elizabeth Stewart, married 1380 David Lindsay, 1st Earl of Crawford
Egidia Stewart, married 1387 Sir William Douglas of Nithsdale
The confusion about the circumstances of his first marriage would later lead to conflict amongst the descendants of his first marriage (which included James I of Scotland) and the unquestionably legitimate descendants of his second marriage.

Robert had also eight illegitimate children: By Moira Leitch:

John Stewart of Bute, Sheriff of Bute, "the Black Stewart"
By Marion Cardny:

Alexander Stewart, Canon of Glasgow
John Stewart of Dundonald, "the Red Stewart"
James Stewart of Kinfauns
Walter Stewart
John Stewart of Burley, Lord of Burley
Sir John Stewart of Cairdney
Sir Alexander Stewart of Inverlunan
Thomas Stewart, Dean of Dunkeld, Archdeacon and Bishop-elect of St. Andrews

References

Primary
Andrew of Wyntoun, The Orygynale Cronykil of Scotland, edited by D. Laing (Edinburgh, 1872-1879)
John of Fordun, Scotichronicon, continued by Walter Bower, edited by Thomas Hearne (Oxford, 1722)
John Major, Historia majoris Britanniae, translated by A Constable (Edinburgh, 1892)

Secondary
Tytler, PF, History of Scotland (Edinburgh, 1841-1843).