1791 (MDCCXCI) was a common year starting on Saturday of the Gregorian calendar and a common year starting on Wednesday of the Julian calendar, the 1791st year of the Common Era (CE) and Anno Domini (AD) designations, the 791st year of the 2nd millennium, the 91st year of the 18th century, and the 2nd year of the 1790s decade. As of the start of 1791, the Gregorian calendar was 11 days ahead of the Julian calendar, which remained in localized use until 1923.
- January 2 – Big Bottom massacre in the Ohio Country, marking the beginning of the Northwest Indian War.
- January 12 – Holy Roman troops reenter Liège, heralding the end of the Liège Revolution, and the restoration of its Prince-Bishops.
- January 25 – The British Parliament passes the Constitutional Act 1791, splitting the old province of Quebec into Upper and Lower Canada.
- February 8 – The Bank of the United States, based in Philadelphia, is incorporated by the federal government with a 20-year charter and started with $10,000,000 capital. 
- February 21 – The United States opens diplomatic relations with Portugal.
- March 2 – French Revolution:
- March 4 – Vermont is admitted, as the 14th U.S. state.
- March 13 – Thomas Paine's chief work Rights of Man (first part) is published in London.
- March – French Revolution: In France, the National Constituent Assembly accepts the recommendation of its Commission of Weights and Measures, that the nation should adopt the metric system.
- April 21 – The first of forty boundary stones, delineating the borders of the new District of Columbia in the United States, is laid at Jones Point Light, in Alexandria, Virginia.
- May 3 – The Sejm of the Polish–Lithuanian Commonwealth proclaims the Constitution of May 3, 1791, the first modern codified constitution in Europe.
- June 20 – French Revolution – Flight to Varennes: The French Royal Family is captured when they try to flee in disguise.
- June 21 – The Ordnance Survey is founded in Great Britain.
- October – French Revolution: The Legislative Assembly (France) convenes.
- October 9 – Mission Nuestra Señora de la Soledad is founded by Father Fermín Lasuén, becoming the 13th mission in the California mission chain.
- October 28 – French Revolution: The Declaration of the Rights of Woman and the Female Citizen is published in France.
- November 4 – St. Clair's Defeat, the worst loss suffered by the United States Army in fighting against American Indians, takes place in what is now Mercer County, Ohio. Miami fighters led by Chief Mihsihkinaahkwa (Little Turtle) and by Shawnee warriors commanded by War Chief Weyapiersenwah (Blue Jacket) rout the forces of General Arthur St. Clair and kill 630 U.S. soldiers, along with hundreds of civilians. 
- December 4 – The first issue of The Observer, the world's first Sunday newspaper, is published in London.
- December 5 – Austrian composer Wolfgang Amadeus Mozart dies aged 35 at his home in Vienna, perhaps of acute rheumatic fever, and is buried two days later.
- December 15 – Ratification by the states of the first ten amendments to the United States Constitution is completed, creating the United States Bill of Rights. Two additional amendments remain pending, and one of these is finally ratified in 1992, becoming the Twenty-seventh Amendment.
- December 23 – Pale of Settlement established by ukase of Catherine the Great, specifying those areas of the Russian Empire in which Jews are permitted permanent residency.
- The first American ship reaches Japan.
- An ordinance is written barring the game of baseball within 80 yards of the Meeting House in Pittsfield, Massachusetts (first known reference to the game of baseball in North America).
- The first serious secondary education school open to girls in Denmark, the Døtreskolen af 1791, is founded in Copenhagen.
- The School for the Indigent Blind, the oldest continuously operating specialist school of its kind in the world, is founded in Liverpool, England, by blind ex-merchant seaman, writer and abolitionist Edward Rushton.
- The Casbah of Algiers Palace is completed.
- January 15 – Franz Grillparzer, Austrian writer (d. 1872)
- January 28 – Ferdinand Hérold, French composer (d. 1833)
- February 12 – Peter Cooper, American industrialist, inventor and philanthropist (d. 1883)
- February 21
- March 20 – Marie Ellenrieder, German painter (d. 1863)
- April 23 – James Buchanan, 15th President of the United States (d. 1868)
- April 27 – Samuel Morse, American inventor (d. 1872)
- June 1 – John Nelson (lawyer), American lawyer (d. 1860)
- June 30 – Félix Savart, French physicist (d. 1841)
- July 26 – Franz Xaver Wolfgang Mozart, Austrian composer, pianist (d. 1844)
- September 5 – Giacomo Meyerbeer, German composer (d. 1864)
- September 21 – István Széchenyi, Hungarian politician, writer (d. 1860)
- September 22 – Michael Faraday, British scientist (d. 1867)
- September 23
- September 26 – Théodore Géricault, French painter (d. 1824)
- October 29 – John Elliotson, British physician (d. 1868)
- November 11 – Josef Munzinger, member of the Swiss Federal Council (d. 1855)
- December 7 – Ferenc Novák Hungarian Slovene writer (d. 1836)
- December 26 – Charles Babbage, British mathematician, inventor (d. 1871)
- January 11 – William Williams Pantycelyn, Welsh hymnist (b. 1717)
- January 23 – Johann Phillip Fabricius, German missionary (b. 1711)
- March 2 – John Wesley, English founder of Methodism (b. 1703)
- March 10 – William Wentworth, 2nd Earl of Strafford (1722–1791), England (b. 1722)
- March 14 – Johann Salomo Semler, German historian, Bible commentator (b. 1725)
- March 31 – Ralph Verney, 2nd Earl Verney of Ireland (b. 1714)
- April 2 – Honoré Gabriel Riqueti, comte de Mirabeau, French revolutionary leader (b. 1749)
- April 19 – Richard Price, Welsh philosopher (b. 1723)
- April 24 – Benjamin Harrison V, signer of the American Declaration of Independence (b. 1726)
- May 9 – Francis Hopkinson, American signer of the Declaration of Independence (b. 1737)
- June 5 – Frederick Haldimand, Swiss-born British colonial governor (b. 1718)
- June 10 – Toussaint-Guillaume Picquet de la Motte, French admiral (b. 1720)
- June 17 – Selina Hastings, Countess of Huntingdon, English Methodist leader (b. 1707)
- June 30 – Jean-Baptiste Descamps, Flemish painter and art historian (b. 1714)
- July 9 – Jacques-Nicolas Tardieu, French engraver (b. 1716)
- July 17 – Martin Dobrizhoffer, Austrian Jesuit missionary (b. 1717)
- July 25 – Isaac Low, American delegate to the Continental Congress (b. 1735)
- August 22 – Johann David Michaelis, German biblical scholar and teacher (b. 1717)
- September 25 – William Bradford, American printer (b. 1719)
- October 7 – Mary Frances of the Five Wounds, Italian Franciscan saint (b. 1715)
- October 12
- October 16 – Grigory Potemkin, Russian military leader, statesman, nobleman and favourite of Catherine the Great (b. 1739)
- November 4 – Richard Butler, American soldier (b. 1743)
- November 16 – Edward Penny, British painter (b. 1714)
- December 5 – Wolfgang Amadeus Mozart, Austrian composer (b. 1756)
- December 12
- December 13 – Mathieu Tillet, French botanist (b. 1714)
- December 19 – Jean-François de Neufforge, Flemish architect and engraver (b. 1714)
- December 27 – John Monro, British physician of Bethlem Hospital (b. 1716)
- date unknown – Maria Petraccini, Italian anatomist, physician (b. 1759)
- Harper's Encyclopaedia of United States History from 458 A. D. to 1909, ed. by Benson John Lossing and, Woodrow Wilson (Harper & Brothers, 1910) p169
- The Hutchinson Factfinder. Helicon. 1999. ISBN 1-85986-000-1.
- "A short history of the Ordnance Survey of Great Britain" (PDF).
- Robert M. Owens, Red Dreams, White Nightmares: Pan-Indian Alliances in the Anglo-American Mind, 1763–1815 (University of Oklahoma Press, 2015)
- "Interior of Governors Palace, Algiers, Algeria". World Digital Library. 1899. Retrieved 2013-09-25.
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