January 17 – Tuscarora War: Colonel James Moore leads the Carolina militia out of Albemarle County, North Carolina in a second offensive against the Tuscarora. Heavy snows force the troops to take refuge in Fort Reading on the Pamlico River.
February 1 – Skirmish at Bender
February 4 – Tuscarora War: The Carolina militia under Colonel James Moore leaves Fort Reading to continue the campaign against the Tuscarora.
February 25 – Frederick William I of Prussia begins his reign.
March 1 – Tuscarora War: Colonel James Moore's Carolina militia lays siege to the Tuscaroran stronghold of Fort Neoheroka, located a few miles up Contentnea Creek from Fort Hancock.
March 20 – Tuscarora War: Colonel James Moore's Carolina militia launches a major offensive against Fort Neoheroka.
March 23 – Tuscarora War: Fort Neoheroka falls to the Carolina militia, effectively ending the Tuscarora nation's military strength. Two Tuscaroran allies, the Machapunga and Coree tribes, continue offensive actions against North Carolina.
March 27 – First Treaty of Utrecht between Great Britain and Spain. Philip V accepted by Britain and Austria as king of Spain; Spain cedes Gibraltar and Minorca to Britain.
April 11 – Second Treaty of Utrecht between Britain and France ends the War of the Spanish Succession. France cedes Newfoundland, Acadia, Hudson Bay and St Kitts to Britain.
April 14 – First performance, in London, of Joseph Addison's libertarian play Cato, a Tragedy, which will be influential on both sides of the Atlantic.
April 19 – With no living male heirs, Charles VI, Holy Roman Emperor, issues the Pragmatic Sanction of 1713 to ensure one of his daughters will inherit the Habsburg lands.
June 1 (approx.) – Tuscarora War: Colonel James Moore leads the Carolina militia into the Pamlico Peninsula to defeat the Machapunga and Coree tribes.
June 23 – French residents of Acadia given one year to declare allegiance to Great Britain or leave Nova Scotia.
July 13 – The Treaty of Portsmouth brings an end to Queen Anne's War.
September 1 – Tuscarora War: The Carolina militia led by Colonel James Moore returns to South Carolina after mixed success in the campaign against the Machapunga and Coree.
Ars Conjectandi, a seminal work on probability by Jacob Bernoulli is published eight years after his death, by his nephew, Niklaus Bernoulli.
January 2 – Marie Dumesnil, French actor (d. 1803)
January 5 – Jorge Juan y Santacilia, Geodesist (d. 1773)
January 7 – Giovanni Battista Locatelli, Italian opera director (d. 1785)
January 13 – Charlotte Charke, British actor and writer (d. 1760)
January 17 – Jean Chrétien Fischer, French general (d. 1762)
January 22 – Marc-Antoine Laugier, Jesuit priest and architectural theorist (d. 1769)
Anthony Benezet, French-born American abolitionist and educator who was active in Philadelphia (d. 1784)
Adam Drummond, British politician (d. 1786)
John Drummond, 10th of Lennoch, British politician (d. 1752)
February 2 – Maria Margarida de Lorena, 2nd Duchess of Abrantes, Portuguese noble, court lady (d. 1780)
February 11 – Diane Adélaïde de Mailly, third of the five famous de Nesle sisters (d. 1760)
February 13 – Domènec Terradellas, Spanish opera composer (d. 1751)
February 20 – Anna Maria Elvia, Swedish feminist writer (d. 1784)
Edward Cornwallis, British military officer who was the first Governor of Nova Scotia (d. 1776)
Frederick Cornwallis, Archbishop of Canterbury (d. 1783)
March 8 – Gian Carlo Passeroni, Italian writer (d. 1803)
March 12 – Johann Adolph Hass, Clavichord maker (d. 1771)
March 17 – Sir Charles Asgill, 1st Baronet, British politician (d. 1788)
March 21 – Francis Lewis, signer of the United States Declaration of Independence as a representative of New York (d. 1803)
March 23 – Bowen Southwell, Irish politician (d. 1796)
March 26 – Peter Oliver (loyalist), Massachusetts colonial judge (d. 1791)
March 28 – Juan Nentvig, German anthropologist (d. 1768)
March 29 – John Ponsonby (politician), Irish politician (d. 1789)
April 7 – Nicola Sala, Italian opera composer (d. 1801)
April 10 – John Whitehurst, English clockmaker (d. 1788)
April 11 – Luise Gottsched, German poet, playwright, essayist and translator (d. 1762)
April 12 – Guillaume Thomas François Raynal, French writer and man of letters during the Age of Enlightenment (d. 1796)
April 13 – Pierre Jélyotte, French operatic tenor (d. 1797)
April 17 – Samuel Graves, British Royal Navy admiral, known for his role early in the American War of Independence (d. 1787)
Anna Maria Hilfeling, Swedish artist (d. 1783)
Louis de Noailles, French peer and Marshal of France (d. 1793)
April 22 – Peter Du Cane, Sr., British businessman (d. 1803)
May 6 – Charles Batteux, French philosopher and writer on aesthetics (d. 1780)
May 7 – Charles Townley (officer of arms), Officer of Arms (d. 1774)
May 11 – James Drummond, 3rd Duke of Perth, British noble (d. 1747)
Alexis Clairaut, prominent French mathematician (d. 1765)
Louis François de Monteynard, French soldier and statesman (d. 1791)
József Károly Hell, Hungarian mining engineer (d. 1789)
Edward Wortley Montagu (traveller), English author and traveller (d. 1776)
Andrzej Mokronowski, Polish general (d. 1784)
John Stuart, 3rd Earl of Bute, Prime Minister of Great Britain (1762–1763) (d. 1792)
May 31 – Giuseppe Maria Buonaparte, Corsican politician (d. 1763)
June 3 – Robert Petre, 8th Baron Petre, renowned horticulturist and a British peer (d. 1742)
June 10 – Princess Caroline of Great Britain, fourth child and third daughter of George II (d. 1757)
John Allen, 3rd Viscount Allen, Irish politician (d. 1745)
Edward Capell, English Shakespearian critic (d. 1781)
June 16 – Meshech Weare, First Governor of New Hampshire (d. 1786)
June 20 – Georg Anton Urlaub, German painter (d. 1759)
June 22 – Lord John Sackville, second son of Lionel Sackville (d. 1765)
July 1 – Benjamin Green (merchant), Canadian mercahnt and judge (d. 1772)
Stanhope Aspinwall, British diplomat (d. 1771)
Jean Godin des Odonais, French cartographer and naturalist (d. 1792)
July 9 – John Newbery, English publisher and bookseller (d. 1767)
July 10 – Anna Rosina de Gasc, German portrait painter (d. 1783)
July 18 – Gaetano Matteo Pisoni, Swiss-Italian architect (d. 1782)
July 22 – Jacques-Germain Soufflot, French architect in the international circle that introduced neoclassicism (d. 1780)
July 27 – Sophie Charlotte of Brandenburg-Bayreuth, German noble (d. 1747)
August 1 – Charles I, Duke of Brunswick-Wolfenbüttel (d. 1780)
Bernardo de Miera y Pacheco, Spanish cartographer (d. 1785)
Princess Elisabeth Albertine of Saxe-Hildburghausen, Duchess consort of Mecklenburg-Strelitz (d. 1761)
August 6 – Marie Sophie de Courcillon, French noblewoman and Duchess of Rohan-Rohan, Princess of Soubise by marriage (d. 1756)
August 11 – Lebbeus Harris, Canadian politician (d. 1792)
August 17 – Antoine de Montazet, French archbishop (d. 1788)
August 25 – Vijaya Raghunatha Raya Tondaiman I, Raja of Pudukkottai (d. 1769)
August 27 – Anton August Beck, German engraver (d. 1787)
September 3 – Jean Baptiste de La Vérendrye, eldest son of Pierre Gaultier de Varennes et de La Vérendrye and Marie-Anne Dandonneau Du Sablé (d. 1736)
Gowin Knight, British physicist (d. 1772)
John Needham, British biologist and priest (d. 1781)
Giuseppe Maria Buondelmonti, Italian philosopher (d. 1757)
Charles Lucas, Irish politician (d. 1771)
September 14 – Johann Kies, German astronomer and mathematician (d. 1781)
September 16 – Charles Lucas (politician), Irish apothecary (d. 1771)
September 23 – Ferdinand VI of Spain, King of Spain (d. 1759)
October 3 – Antoine Dauvergne, French composer and violinist (d. 1797)
October 5 – Denis Diderot, French philosopher (d. 1784)
October 7 – Granville Elliott, Army General, British military expert, working for Britain and Palatine forces (d. 1759)
October 8 – Yechezkel Landau, influential authority in halakha (Jewish law) (d. 1793)
October 12 – Khawaja Muhammad Zaman of Luari, Sufi poet (d. 1775)
Allan Ramsay (artist), prominent Scottish portrait-painter (d. 1784)
Jacques de Romas, French physicist (d. 1776)
Benjamin Andrew, American politician (d. 1790)
James Cecil, 6th Earl of Salisbury, English Earl (d. 1780)
Joseph Redlhamer, Austrian physicist (d. 1761)
October 23 – Pieter Burman the Younger, Dutch academic (d. 1788)
October 24 – Marie Fel, French opera singer (d. 1794)
October 30 – Giuseppe Antonio Landi, Italian painter (d. 1791)
November 1 – Antonio Genovesi, Italian economist (d. 1769)
November 5 – Gorges Lowther (1713–1792), Member of Irish House of Commons (d. 1792)
November 6 – Thomas Osborne, 4th Duke of Leeds, British politician (d. 1789)
Junípero Serra, Christian missionary (d. 1784)
Laurence Sterne, Anglo-Irish novelist and an Anglican clergyman (d. 1768)
November 30 – Johann Balthasar Bullinger, Swiss artist (d. 1793)
December 4 – Gasparo Gozzi, Venetian critic and dramatist (d. 1786)
December 10 – Johann Nicolaus Mempel, German composer and musician (d. 1747)
December 13 – John Baptist Caryll, third Jacobite Baron Caryll of Durford (d. 1788)
December 14 – Martin Knutzen, German philosopher (d. 1751)
December 15 – Welbore Ellis, 1st Baron Mendip, British politician (d. 1802)
December 23 – Maruyama Gondazaemon, Sumo wrestler (d. 1749)
December 27 – Giovanni Battista Borra, Italian architect and engineer (d. 1770)
December 29 – Nicolas Louis de Lacaille, French astronomer (d. 1762)
January 8 – Arcangelo Corelli, Italian composer (b. 1653)
January 11 – Pierre Jurieu, French Protestant leader (b. 1637)
January 12 – John Vaughan, 3rd Earl of Carbery, Governor of Jamaica and President of the Royal Society (b. 1639)
February 4 – Anthony Ashley-Cooper, 3rd Earl of Shaftesbury, English politician and philosopher (b. 1671)
February 25 – King Frederick I of Prussia (b. 1657)
March 18 – Juraj Jánošík, the Slovak Robin Hood (executed)
May 20 – Thomas Sprat, English minister (b. 1635)
July 7 – Henry Compton, Bishop of Oxford and privy councillor (b. 1632)
October 15 – Johann Michael Feuchtmayer the Elder, artist (b. 1666)
October 20 – Archibald Pitcairne, Scottish physician (b. 1652)
November 7 – Elizabeth Barry, English actress (b. 1658)
November 17 – Abraham van Riebeeck, Governor-General of the Dutch East Indies (b. 1653)
December 14 – Thomas Rymer, English historian (b. 1641)
date unknown – Thomas Ellwood, English religious writer (b. 1639)
1713 (MDCCXIII) was a common year starting on Sunday (dominical letter A) of the Gregorian calendar and a common year starting on Thursday (dominical letter D) of the Julian calendar, the 1713th year of the Common Era (CE) and Anno Domini (AD) designations, the 713th year of the 2nd millennium, the 13th year of the 18th century, and the 4th year of the 1710s decade. As of the start of 1713, the Gregorian calendar was 11 days ahead of the Julian calendar, which remained in localized use until 1923.