1871

January–March

 * January 18 – The member states of the North German Confederation and the south German states unite into a single nation state known as the German Empire. The King of Prussia is declared the first German Emperor as Wilhelm I of Germany in the Hall of Mirrors at the Palace of Versailles.
 * January 21 – Giuseppe Garibaldi's group of French and Italian volunteer troops in support of the French Third Republic win a battle against the Prussians in Dijon.
 * February 9 – United States Commission on Fish and Fisheries is founded.
 * February 21– 41st United States Congress passes "An Act To Provide A Government For The District Of Columbia", also known as the Act of 1871, declaring the government of the District of Columbia a municipal corporation not inconsistent with the Constitution and laws of the United States and the provisions of this act.
 * March 7 – José Paranhos, Viscount of Rio Branco, becomes Prime Minister of the Empire of Brazil, serving for 4 years.
 * March 21 – John Campbell, Marquess of Lorne (whose father, the 8th Duke of Argyll, is the serving Secretary of State for India) marries Princess Louise.
 * March 21 – Otto von Bismarck becomes first Chancellor of the German Empire.
 * March 22
 * In North Carolina, William Holden becomes the first governor of a U.S. state (North Carolina) to be removed from office by impeachment.
 * The United States Army issues an order for the abandonment of Fort Kearny, Nebraska.
 * March 26 – The Paris Commune is formally established in Paris.
 * March 27 – The first rugby union International results in a 4–1 win by Scotland over England.
 * March 29
 * First Surgeon General of the United States (John Maynard Woodworth) appointed.
 * The Royal Albert Hall in London is opened by Queen Victoria.

April–June

 * April – The Stockholms Handelsbank is founded.
 * April 4 – New Jersey Detective Agency chartered. New Jersey State Detectives initiated.
 * April 20 – U.S. President Ulysses S. Grant signs the Civil Rights Act.
 * April 24 – Murder of servant girl Jane Clouson in Eltham.
 * May 4 – The first supposedly Major League Baseball game is played.
 * May 8 – The first Major League Baseball home run is hit by Ezra Sutton of the Cleveland Forest Citys.
 * May 10 – Treaty of Frankfurt is signed confirming the frontiers between Germany and France.
 * May 11 – The first trial in the Tichborne case begins in the London Court of Common Pleas.
 * May 21 – Opening of the first rack railway in Europe, the Vitznau–Rigi Railway on Mount Rigi in Switzerland.
 * May 28 – Following the invasion of the Paris Commune by Government troops, 147 Communards, the last defenders of the workers' district of Belleville, are shot, on the last day of the "Bloody Week" (Semaine Sanglante) in which the Commune is crushed.
 * May 30 – French Third Republic: Government suppression of the Paris Commune rebellion is completed.
 * June 1 – Bombardment of the Selee River Forts: Koreans attack two United States Navy warships.
 * June 10 – United States expedition to Korea: Captain McLane Tilton leads 109 members of the United States Marine Corps in a punitive naval attack on the Han River forts on Ganghwa Island in Korea.
 * June 18 – The University Tests Act removes restrictions limiting access to Oxford, Cambridge and Durham universities to members of the Church of England.

July–September

 * July 20
 * British Columbia joins the confederation of Canada.
 * C. W. Alcock proposes that "a Challenge Cup should be established in connection with the Association", giving birth to the FA Cup.
 * July 21 – August 26 – First ever photographs of Yellowstone National Park region taken by the photographer William Henry Jackson during the Hayden Geological Survey of 1871.
 * July 22 – The foundation stone of the first Tay Rail Bridge is laid; the bridge collapses in a storm eight years later.
 * July 28 – The Annie, the first boat ever launched on Yellowstone Lake in Yellowstone National Park region.
 * August 29 – The abolition of the han system is carried out in Japan.
 * August 31 – Adolphe Thiers becomes the President of the French Republic.
 * September 2 – Whaling Disaster of 1871: The Comet, a brig used by whalers, becomes the first of 33 ships to be crushed in the Arctic ice by an early freeze. Remarkably, all 1,219 people on the abandoned ships are rescued without a single loss of life.
 * September 3 – New York City residents, tired of the corruption of Tammany Hall and Boss Tweed, meet to form the "Committee of Seventy" to reform local politics.

October–December

 * October 8 – Four major fires break out on the shores of Lake Michigan in Chicago, Peshtigo, Wisconsin, Holland, Michigan, and Manistee, Michigan. The Great Chicago Fire is the most famous of these, leaving nearly 100,000 people homeless, although the Peshtigo Fire kills as many as 2,500 people, making it the deadliest fire in United States history.
 * October 12 – Criminal Tribes Act (CTA) enacted by the British Raj in India, naming over 160 communities as "Denotified Tribes", allegedly habitually criminal. It will be repealed in 1949, after Indian independence.
 * October 20 – The Royal Regiment of Artillery forms the first regular Canadian army units when they create two batteries of garrison artillery, which later become the Royal Canadian Artillery.
 * October 24 – 18 Chinese immigrants in Los Angeles' Chinatown are killed by a mob of 500 men in what became known as the Chinese massacre of 1871.
 * October 27
 * Henri, Count of Chambord, refuses to be crowned "King Henry V of France" until France abandons its tricolor and returns to the old Bourbon flag.
 * Boss Tweed, the Grand Sachem of Tammany Hall, is arrested.
 * November 5 – Wickenburg Massacre: six men travelling by stagecoach in Arizona Territory are reportedly murdered by Yavapai people.
 * November 7 – The London–Australia telegraph cable is brought ashore at Darwin.
 * November 10 – Henry Morton Stanley, Welsh-born correspondent for the New York Herald, locates the missing Scottish explorer and missionary Dr. David Livingstone in Ujiji, near Lake Tanganyika, and greets him by saying "Dr. Livingstone, I presume?"
 * November 17
 * The National Rifle Association is granted a charter by the state of New York.
 * George Biddell Airy presents his discovery that astronomical aberration is independent of the local medium.
 * December 10 – German chancellor Otto von Bismarck tries to ban Catholics from the political stage by introducing harsh laws concerning the separation of church and state.
 * December 19 – The city of Birmingham, Alabama, is incorporated with the merger of three existing towns.
 * December 24 – Aida opens in Cairo, Egypt.
 * December 25 – The Reading Football Club is formed.
 * December 26 – Thespis, the first of the Gilbert and Sullivan operas, premières. It does modestly well, but the two composers will not collaborate again for four years.

Date unknown

 * The provinces of Alsace and Lorraine are transferred from France to Germany.
 * British trade unions are legalized.
 * Heinrich Schliemann begins the excavation of Troy.
 * Japan forms its own nationwide police force based on the French model.
 * William M. Tweed serves his last year as the "Boss" of the Tammany Hall political machine in New York.
 * The South Improvement Company is formed by John D. Rockefeller and a group of major railroad interests in an early effort to organize and control the petroleum industry in the USA.
 * The Harvard Summer School is founded.
 * The Constitution of the German Empire abolishes all restrictions on Jewish marriage, choice of occupation, place of residence, and property ownership. Exclusion from government employment and discrimination in social relations remain in effect.
 * The American minister to China takes five warships to attempt to "open up" Korea, but his forces leave after exchanges of fire result in 250 Koreans dying and the Korean government still unwilling to make any concessions.
 * Virginia adopts a new Constitution, taking into account, among other things, all of the counties that had left Virginia in 1863 to form the new non-slave state of West Virginia. No other state has ever formed by breaking off from another without the consent of the legislature of the parent state, as in the cases of Vermont, Kentucky, and Maine.
 * In Hanover, German company Continental AG is founded.

January–June

 * January 7 – Félix Édouard Justin Émile Borel, French mathematician and politician (d. 1956)
 * January 17 – David Beatty, 1st Earl Beatty, British admiral (d. 1936)
 * January 30 – Wilfred Lucas, Canadian-born actor (d. 1940)
 * February 4 – Friedrich Ebert, President of Germany (d. 1925)
 * February 9 – Howard Taylor Ricketts, American pathologist (d. 1910)
 * February 18 – Harry Brearley, English inventor (d. 1948)
 * February 28 – Manuel Díaz Rodríguez, Venezuelan writer (d. 1927)
 * March 1 – Ben Harney, American composer and pianist (d. 1938)
 * March 4 – Boris Galerkin, Russian mathematician (d. 1945)
 * March 5 – Rosa Luxemburg, German politician (d. 1919)
 * March 17 – Konstantinos Pallis, Greek general (d. 1941)
 * March 19 – Schofield Haigh, English cricketer (d. 1921)
 * March 27 – Heinrich Mann, German writer (d. 1950)
 * March 31 – Arthur Griffith, President of Ireland (d. 1922)
 * April 4 – Luke McNamee, American admiral (d. 1952)
 * April 8 – Clarence Hudson White, American photographer (d. 1925)
 * April 15 – Jonathan Zenneck, German physicist and electrical engineer (1959)
 * May 2 – Francis P. Duffy, Canadian-born Roman Catholic priest in the United States (d. 1932)
 * May 3 – Walter Robinson Parr, English-born Congregational pastor in the United States (d. 1922)
 * May 6
 * Victor Grignard, French chemist, Nobel Prize in Chemistry laureate (d. 1935)
 * Christian Morgenstern, German author (d. 1914)
 * May 15 – Kōzō Satō, Japanese admiral (d. 1948)
 * May 27 – Georges Rouault, French painter and graphic artist (d. 1958)
 * June 12 – Ernst Stromer, German paleontologist (d. 1952)
 * June 14 – Jacob Ellehammer, Danish inventor (d. 1946)
 * June 17 – James Weldon Johnson, American author, politician, diplomat, critic, journalist, poet, anthologist, educator, lawyer, songwriter and early civil rights activist (d. 1938)
 * June 23 – Jantina Tammes, Dutch plant biologist (d. 1947)
 * June 26 – Reginald R. Belknap, United States Navy rear admiral (d. 1959)

July–December

 * July 10 – Marcel Proust, French writer (d. 1922)
 * July 17 – Lyonel Feininger, German painter (d. 1956)
 * July 18 – Sada Yacco, Japanese stage actress (d. 1946)
 * July 25 – Richard Ernest Turner, Canadian soldier (d. 1961)
 * August 1 – John Lester, American cricketer (d. 1969)
 * August 10 – Aino Sibelius, wife of composer Jean Sibelius (d. 1969)
 * August 13 – Karl Liebknecht, German politician (d. 1919)
 * August 14 – Guangxu Emperor of China (d. 1908)
 * August 19
 * Orville Wright, American aviation pioneer, co-inventor of the airplane with brother Wilbur (d. 1948)
 * Joseph E. Widener, American art collector (d. 1943)
 * August 25 – Ross Winn, American anarchist writer and publisher (d. 1912)
 * August 27 – Theodore Dreiser, American writer (d. 1945)
 * August 29 – Albert François Lebrun, French politician (d. 1950)
 * August 30 – Ernest Rutherford, New Zealand physicist, recipient of the Nobel Prize in Chemistry (d. 1937)
 * September 1 – J. Reuben Clark, Under Secretary of State for U.S. President Calvin Coolidge (d. 1961)
 * September 10 – Charles Collett, English Great Western Railway chief mechanical engineer (d. 1952)
 * September 17 – Eivind Astrup, Norwegian Arctic explorer (d. 1895)
 * September 19 – Frederick Ruple, Swiss-born American portrait painter (d. 1938)
 * September 24 – Lottie Dod, English athlete (d. 1960)
 * September 26 – Winsor McCay, American cartoonist and animator (d. 1934)
 * September 27 – Grazia Deledda, Italian writer, Nobel Prize laureate (d. 1936)
 * September 28 – Pietro Badoglio, Italian general and prime minister (d. 1956)
 * October 2 – Cordell Hull, United States Secretary of State, recipient of the Nobel Peace Prize (d. 1955)
 * October 19 – Walter Bradford Cannon, American physiologist (d. 1945)
 * October 11
 * Harriet Boyd Hawes, American archaeologist (d. 1945)
 * Johan Oscar Smith, Norwegian Christian leader, founder of Brunstad Christian Church (d. 1943)
 * October 25 – John Gough, British general and Victoria Cross recipient (d. 1915)
 * October 30
 * Paul Valéry, French poet (d. 1945)
 * Buck Freeman, American baseball player (d. 1949)
 * November 1 – Stephen Crane, American writer (d. 1900)
 * November 3 – Albert Goldthorpe, English rugby league footballer (d. 1943)
 * December 9 – Joe Kelley, American Baseball Hall of Famer (d. 1943)
 * December 13 – Emily Carr, Canadian artist (d. 1945)
 * date unknown - Sevasti Qiriazi, Albanian educator and women's rights activist (d. 1949)
 * date unknown – Qiu Yufang, Chinese revolutionary, writer and feminist (d. 1904)

January–June

 * January 8 – José Trinidad Cabañas, a Honduran General, President and National Hero (b. 1805)
 * January 13 – Kawakami Gensai, a highly skilled swordsman and one of the four most notable assassins of the Bakumatsu period. In the manga Rurouni Kenshin, the main character is a skilled swordsman named Kenshin Himura.  He is loosely based on Gensai.
 * January 15 – Edward C. Delavan, American temperance movement leader (b. 1793)
 * January 19 – Sir William Denison, Governor of New South Wales (b. 1804)
 * January 25 – Jeanne Villepreux-Power, French marine biologist, (b. 1794)
 * February 12 – Alice Cary, American poet, sister to Phoebe Cary (1824–1871) (b. 1820)
 * February 20 – Paul Kane, Irish-born painter (b. 1810)
 * March – Emma Fürstenhoff, Swedish florist (b. 1802)
 * March 18 – Augustus De Morgan, professor of mathematics and mathematician (b. 1806)
 * April 7
 * Prince Alexander John of Wales (b. April 6, prematurely)
 * April 7 – Wilhelm von Tegetthoff, Austrian admiral (b. 1827)
 * April 25 – Jane Clouson, teenaged British murder victim (b. 1854)
 * May 11 – John Herschel, English astronomer (b. 1792)
 * May 12 – Elzéar-Henri Juchereau Duchesnay, Canadian politician (b. 1809)
 * May 23 – Jarosław Dąbrowski, Polish general (b. 1836)
 * June 9 – Anna Atkins, British botanist (b. 1799)

July–December

 * July 5 – Cristina Trivulzio Belgiojoso, Italian noble, patriot, writer and journalist  (b. 1808)
 * July 15 – Tad Lincoln, Youngest son of American President Abraham Lincoln (b. 1853)
 * July 31 – Phoebe Cary, American poet, sister to Alice Cary (1820–1871) (b. 1824)
 * August 9 – John Paterson, politician in the New South Wales Legislative Assembly (b. 1831)
 * September 16 – Jan Erazim Vocel, Czech poet, archaeologist, historian and cultural revivalist (b. 1803)
 * September 20 – John Coleridge Patteson, Anglican bishop and missionary (martyred) (b. 1827)
 * September 23 – Louis-Joseph Papineau, Canadian politician (b. 1786)
 * October 18 – Charles Babbage, English mathematician and inventor (b. 1791)
 * November 22 – Oscar James Dunn, Lieutenant Governor of Louisiana (b. 1825)
 * December 21 – Luise Aston, German author and feminist (b. 1814)
 * December 28 – John Henry Pratt, English clergyman and mathematician (b. 1809)
 * Constance Trotti, Belgian salonniére and culture patron (b. 1800)