This article is about the particular significance of the year 1791 to Wales and its people.

• Lord Lieutenant of Anglesey - Henry Paget^[1][2][3][4]
• Lord Lieutenant of Brecknockshire and Monmouthshire – Henry Somerset, 5th Duke of Beaufort^[5]
• Lord Lieutenant of Caernarvonshire - Thomas Bulkeley, 7th Viscount Bulkeley^[6]
• Lord Lieutenant of Cardiganshire – Wilmot Vaughan, 1st Earl of Lisburne^[2]
• Lord Lieutenant of Carmarthenshire – John Vaughan
• Lord Lieutenant of Denbighshire - Richard Myddelton
• Lord Lieutenant of Flintshire - Sir Roger Mostyn, 5th Baronet
• Lord Lieutenant of Glamorgan – John Stuart, Lord Mountstuart^[7]
• Lord Lieutenant of Merionethshire - Watkin Williams^[8]
• Lord Lieutenant of Montgomeryshire – George Herbert, 2nd Earl of Powis^[2]
• Lord Lieutenant of Pembrokeshire – Richard Philipps, 1st Baron Milford^[2][9]
• Lord Lieutenant of Radnorshire – Thomas Harley (politician, born 1730) (from 8 April)^[10][2]
• Bishop of Bangor – John Warren^[11][12]
• Bishop of Llandaff – Richard Watson^[13]
• Bishop of St Asaph – Lewis Bagot^[14][11]
• Bishop of St Davids – Samuel Horsley^[15]

• June - William Jones distributes copies of an address at the Llanrwst eisteddfod, titled To all Indigenous Cambro-Britons, calling on poor Welsh farmers to set up a colony in the United States.^[16]
• Richard Phillips builds Clyne Castle.^[17]
• Peter Williams is excommunicated by the Methodists for publishing Sabellian heresy.
• Probable date of completion of Methodist chapel at Earlswood, Monmouthshire.
• Thomas Jones becomes High Sheriff of Radnorshire.

• 23 February - Sir John Cowell-Stepney, baronet, landowner and politician (d. 1877)^[18]
• 5 December - William Henry Yelverton, MP for Carmarthen Boroughs 1832–1835 (d. 1884)^[19]
• date unknown
  • Robert Everett, Independent minister and writer (d. 1875)^[20]
  • Thomas Fothergill, ironmaster (d. 1858)

• 11 January - William Williams (Pantycelyn), poet and hymn-writer, 73^[21]
• 13 February - William Parry, artist, 48^[22]
• 19 April - Richard Price, philosopher, 68^[23]
• 17 September - David Morris (hymn writer), 47