|Covid-19|January 2 – Alexander Pope's poem Epistle to Dr Arbuthnot published in London.
January 8 – George Frideric Handel's opera Ariodante is premièred at the Royal Opera House in Covent Garden, London.
February 14 – The Order of St. Anna is established in Russia, in honour of the daughter of Peter the Great.
April 13 – Emperor Sakuramachi accedes to the throne of Japan.
April 16 – Alcina, George Frideric Handel's Italian opera, premieres at the Royal Opera House in Covent Garden, London.
May 22 – George Hadley publishes the first explanation of the trade winds.
June 24 – effective date of Great Britain Witchcraft Act of 1735 which criminalized claimants accusing persons of practising witchcraft or of possessing magical powers.
July 11 – Pluto (not known at this time) enters a fourteen-year period inside the orbit of Neptune, which will not recur until 1979.
August 14 – Freedom of the press: The New York Weekly Journal writer John Peter Zenger is acquitted of seditious libel against the royal governor of New York, on the basis that what he published was true.
October – War of the Polish Succession: A preliminary peace, ratified in 1738, is concluded.
October 18 – The Qianlong Emperor succeeds his father, the Yongzheng Emperor, and begins a 60-year-long reign of the Qing dynasty.
Linnaeus publishes his Systema Naturae.
Russo-Turkish War, 1735-1739: Russian forces fail to occupy the Crimea due to rasputitsa.
A shipbuilding industry begins in Mumbai.
Leonhard Euler solves the Basel problem, first posed by Pietro Mengoli in 1644, and the Seven Bridges of Königsberg problem.
The King's Highway (Charleston to Boston) is completed.
Quebec: Construction begins on the Chemin du roy between Quebec and Montreal.
Augusta, Georgia, is founded.
Cobalt is discovered and isolated by Georg Brandt.
First successful Appendectomy by french surgeon Claudius Aymand in London
Paul Revere, American silversmith and patriot (d. 1818)
Edward Telfair, Scottish-born American politician and Governor of Georgia (d. 1807)
January 8 – John Carroll (priest) first Roman Catholic Archbishop in the U.S. (d. 1815)
January 9 – John Jervis, 1st Earl of St Vincent, British Royal Navy admiral (d. 1823)
January 27 – Étienne Clavière, French financier and politician (d. 1793)
February 28 – Alexandre-Théophile Vandermonde French musician and chemist (d. 1796)
March 1 – Caroline Thielo, Danish actress (d. 1754)
March 29 – Johann Karl August Musäus German author (d. 1787)
April 13 – Isaac Low, New York delegate to the Continental Congress (d. 1791)
May 1 – Lorenzo Hervás y Panduro Spanish Jesuit philologist (d. 1809)
May 23 – Charles Joseph, Prince de Ligne, (d. 1814)
June 16 – Nicolas Bernard Lépicié, French painter (d. 1784)
July 10 – Ulrika Pasch, Swedish painter (d. 1796)
September 5 – Johann Christian Bach, German composer (d. 1782)
September 20 – James Keir, Scottish geologist, chemist, and industrialist (d. 1820)
September 28 – Augustus FitzRoy, 3rd Duke of Grafton, Prime Minister of the United Kingdom (d. 1811)
October 1 – Augustus Henry Fitzroy, 3rd Duke of Grafton (d. 1811)
October 6 – Jesse Ramsden, English astronomer, mathematician and inventor (d. 1800)
October 9 – Karl Wilhelm Ferdinand, Duke of Brunswick (d. 1806)
October 21 – Richard Gough, English antiquary (d. 1809)
October 30 – John Adams, 2nd President of the United States (d. 1826)
November 10 – Granville Sharp, English abolitionist (d. 1813)
December 29 – Thomas Banks, English sculptor and artist (d. 1805)
December 31 – Jean de Crévecoeur, French-American writer (d. 1813)
date unknown – William Bell, English portrait painter from Newcastle upon Tyne (d. c. 1806)
January 12 – John Eccles, British composer (b. 1668)
January 13 – Polyxena of Hesse-Rotenburg, Queen consort of Sardinia (b.1706)
January 18 – Maria Clementina Sobieski, Polish princess (b. 1702)
February 27 – John Arbuthnot, British physician and author (b. 1667)
April 5 – William Derham, English minister and writer (b. 1657)
April 5 – Samuel Wesley, English poet and religious leader (b. 1662)
June 10 – Thomas Hearne, British antiquarian (b. 1678)
July 26 – Jesper Swedberg, Swedish bishop (b. 1653)
September 27 – Peter Artedi, Swedish naturalist (drowned) (b. 1705)
October 8 – Yongzheng Emperor of Qing China (b. 1678)
November 12 – Landon Amadour poet and composer (b. 1714)
December 14 – Thomas Tanner, English bishop and antiquarian (b. 1674)
1735 (MDCCXXXV) was a common year starting on Saturday (dominical letter B) of the Gregorian calendar and a common year starting on Wednesday (dominical letter E) of the Julian calendar, the 1735th year of the Common Era (CE) and Anno Domini (AD) designations, the 735th year of the 2nd millennium, the 35th year of the 18th century, and the 6th year of the 1730s decade. As of the start of 1735, the Gregorian calendar was 11 days ahead of the Julian calendar, which remained in localized use until 1923.