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1745 (MDCCXLV) was a common year starting on Friday of the Gregorian calendar and a common year starting on Tuesday (dominical letter F) of the Julian calendar, the 1745th year of the Common Era (CE) and Anno Domini (AD) designations, the 745th year of the 2nd millennium, the 45th year of the 18th century, and the 6th year of the 1740s decade. As of the start of 1745, the Gregorian calendar was 11 days ahead of the Julian calendar, which remained in localized use until 1923.
- July 9 – War of the Austrian Succession – Battle of Melle: The French are victorious in an engagement against the Pragmatic Allies.
- July 26 – The first recorded women's cricket match takes place in Surrey, England.
- August 19 – The Jacobite rising of 1745 begins at Glenfinnan, Scotland, where Charles Edward Stuart raises his standard.
- September 1 – Catherine the Great marries Peter III of Russia, in Saint Petersburg.
- September 11 – Jacobite rising of 1745: Jacobites enter Edinburgh; six days later, Charles Edward Stuart proclaims his father James Francis Edward Stuart, as James VIII of Scotland.
- September 12 – Francis I is elected Holy Roman Emperor with the support of his wife, Maria Theresa. He is the successor of Charles VII Albert of Bavaria, an enemy of the House of Habsburg, who died on January 20 of this year.
- September 14 – Madame de Pompadour is officially presented, at the court of Louis XV of France.
- September 16 – Jacobite rising of 1745 – "Canter of Coltbrigg": The British 13th and 14th Dragoons flee the Jacobites, near Edinburgh.
- September 21 – Battle of Prestonpans: British Government forces are defeated by the Jacobites in Scotland.
- December 4 – Jacobite rising of 1745: The Scottish Jacobite army reaches as far south as Derby in England, causing panic in London; two days later it begins to retreat.
- December 18 – Jacobite rising of 1745 – Clifton Moor Skirmish: The Jacobites are victorious, in the last action between two military forces on English soil.
- December 23 – Jacobite rising of 1745 – Battle of Inverurie: The Jacobites are victorious over British royal troops.
- December 25 – The Treaty of Dresden gives Prussia full possession of Silesia.
- December 28 – For 5 days, fire destroys buildings in Istanbul.
- January 1 – Anthony Wayne, United States Army officer, statesman, and member of the United States House of Representatives (d. 1796)
- January 6 – Jacques-Étienne Montgolfier, French inventor (d. 1799)
- February 2 – Hannah More, English religious writer, Romantic and philanthropist (d. 1833)
- February 18 – Alessandro Volta, Italian physicist (d. 1827)
- February 20 – Henry James Pye, English poet (d. 1813)
- February 21 – Olof Tempelman, Swedish architect (d. 1816)
- March 4
- March 10 – John Gunby, Maryland soldier in the American Revolutionary War (d. 1807)
- March 25 – John Barry, officer in the Continental Navy during the American Revolutionary War, and later in the United States Navy (d. 1803)
- April 20 – Philippe Pinel, French physician (d. 1826)
- April 29 – Oliver Ellsworth, Chief Justice of the United States (d. 1807)
- July 8 – Sara Banzet, French educator and diarist (d. 1774)
- July 13 – Robert Calder, British naval officer (d. 1818)
- August 20 – Francis Asbury, American Methodist Bishop (d. 1816)
- August 30 – Johann Hieronymus Schröter, German astronomer (d. 1816)
- September 16 – Mikhail Illarionovich Kutuzov, Russian field marshal (d. 1813)
- November 13 – Valentin Haüy, French educator, founder of the first school for the blind (d. 1822)
- November 23 – John Treadwell, the fourth Governor of Connecticut (d. 1823)
- December 2 – Queen Jeongsun, Korean regent (d. 1805)
- December 12 – John Jay, American politician and 1st Chief Justice of the United States (died 1829)
- December 15 – Johann Gottfried Koehler, German astronomer (d. 1801)
- December 24 – William Paterson, American politician and Associate Justice of the Supreme Court of the United States (d. 1806)
- date unknown
- January 20 – Charles VII, Holy Roman Emperor (b. 1697)
- March 18 – Robert Walpole, first Prime Minister of the United Kingdom (b. 1676)
- May 9 – Tomaso Antonio Vitali, Italian violinist and composer (b. 1663)
- May 22 – François-Marie, 1st duc de Broglie, French military leader (b. 1671)
- October 19 – Jonathan Swift, Anglo-Irish writer (b. 1667)
- November 16 – James Butler, 2nd Duke of Ormonde, Irish statesman and soldier (b. 1665)
- December 8 – Étienne Fourmont, French orientalist (b. 1683)
- December 19 – Jean-Baptiste van Loo, French painter (b. 1684)
- date unknown – Hedvig Catharina De la Gardie, Swedish salonniére (b. 1695)