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 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 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.
- July 8 – Composer Joseph Haydn is awarded an honorary doctorate of music, at the University of Oxford.
- July 14 – The Priestley Riots against Dissenters break out in Birmingham, England.
- July 17 – French Revolution: The Champ de Mars Massacre occurs.
- August 4 – The Treaty of Sistova is signed, ending the Ottoman–Habsburg wars.
- August 6 – The Brandenburg Gate in Berlin is finished.
- August 21 – Haitian Revolution: A slave rebellion breaks out in the French colony of Saint-Domingue.
- August 26 – John Fitch is granted a patent for the steamboat in the United States.
- August 27 – Third Anglo-Mysore War – Battle of Tellicherry off the south-west coast of India: A British Royal Navy patrol forces a French convoy bound for Mysore to surrender.
- September 6 – Wolfgang Amadeus Mozart's opera seria, La clemenza di Tito, premières at the Estates Theatre in Prague, to mark the coronation of Leopold II as King of Bohemia.
- September 9 – The capital of the United States, Washington, D.C., is named after President George Washington.
- September 13 – French Revolution: Louis XVI of France accepts the final version of the completed constitution.
- September 14 – French Revolution: The Papal States lose Avignon to Revolutionary France.
- September 25 – Mission Santa Cruz is founded by Father Fermín Lasuén, becoming the 12th mission in the California mission chain.
- September 28 – French Revolution: The law on Jewish emancipation is promulgated in France, the first such legislation in modern Europe.
- September 30 – Wolfgang Amadeus Mozart's singspiel opera The Magic Flute (Die Zauberflöte) premières at the Freihaus-Theater auf der Wieden, in Vienna.
- 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.
- 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.
- 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 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)
- March 2 – John Wesley, English founder of Methodism (b. 1703)
- March 14 – Johann Salomo Semler, German historian, Bible commentator (b. 1725)
- April 2 – Honoré Gabriel Riqueti, comte de Mirabeau, French revolutionary leader (b. 1749)
- April 19 – Richard Price, Welsh philosopher (b. 1723)
- 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)
- July 17 – Martin Dobrizhoffer, Austrian Jesuit missionary (b. 1717)
- July 25 – Isaac Low, American delegate to the Continental Congress (b. 1735)
- September 25 – William Bradford, American printer (b. 1719)
- November 4 – Richard Butler, American soldier (b. 1743)
- December 5 – Wolfgang Amadeus Mozart, Austrian composer (b. 1756)
- December 12 – Etteilla, French occult cartomancer (b. 1738)
- Date unknown – Maria Petraccini, Italian anatomist, physician (b. 1759)