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 25 – The British Parliament passes the Constitutional Act of 1791, splitting the province of Quebec (a colony in North America spreading from the Gulf of Mexico coast to Hudson Bay)
• March 2 – A semaphore machine is revealed to the public in Paris.
• March 4 – Vermont becomes a state in the United States.
• May 3 – The Polish-Lithuanian Commonwealth Sejm (Parliament) proclaims the Constitution of third May, the first modern codified, or legal code-like, constitution in Europe.
• July 14 – The Priestley Riots happen in Birmingham, West Midlands, England.
• June 20 – The French Royal Family is captured when they try to flee in disguise.
• July 17 – The Champ de Mars Massacre occurs during the French Revolution.
• August 4 – The Treaty of Sistova is signed, ending the Ottoman-Habsburg wars.
• August 6 – Brandenburg Gate in Berlin is finished.
• August 26 – John Fitch is given a patent for the steamboat in the United States.
• September 25 – The Mission Santa Cruz is founded by Father Fermín Francisco de Lasuén, becoming the twelfth mission in the California mission chain.
• September 30 – First showing of Wolfgang Amadeus Mozart's singspiel, a type of opera, Die Zauberflöte (Magic Flute) in the Freihaustheater in Vienna.
• October 9 – The Mission Nuestra Señora de la Soledad, was founded by Father Fermín Francisco de Lasuén, becoming the thirteenth mission in the California mission chain.
• December 4 – The first issue of The Observer, the world's first Sunday newspaper, is published.
• December 15 – The states approve ten amendments to the United States Constitution, which become the Bill of Rights. Two additional amendments are still being thought about, and one of these is finally approved in 1992, becoming the Twenty-seventh Amendment.

• The first American ship reaches Japan.
• Slave rebellion Haiti has begun.

• January 15 – Franz Grillparzer, Austrian writer (d. 1872)
• January 28 – Louis Joseph Ferdinand Herold, French composer (d. 1833)
• February 12 – Peter Cooper, American Industrialist, inventor and philanthropist (d. 1883)
• February 21
  • Carl Czerny, Austrian composer (d. 1857)
  • John Mercer, chemist and industrialist (d. 1866)
• April 23 – James Buchanan, 15th President of the United States (d. 1868)
• April 27 – Samuel Morse, American inventor (d. 1872)
• July 26 – Franz Xaver Wolfgang Mozart, Austrian composer and pianist (d. 1844)
• September 5 – Giacomo Meyerbeer, German composer (d. 1864)
• September 21 – István Széchenyi, Hungarian politician and writer (d. 1860)
• September 22 – Michael Faraday, British scientist (d. 1867)
• September 26 – Théodore Géricault, French painter (d. 1824)
• November 11 – Josef Munzinger, member of the Swiss Federal Council (d. 1855)
• December 26 – Charles Babbage, British mathematician and 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 and Bible commentator (b. 1725)
• 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)
• August 16 – Charles-François de Broglie, marquis de Ruffec, French soldier and diplomat (b. 1719)
• November 4 – Richard Butler (general), American soldier (b. 1743)
• September 25 – William Bradford, American printer (b. 1719)
• December 5 – Wolfgang Amadeus Mozart, Austrian composer (b. 1756)