January 29 – Anna Ivanovna (Anna of Russia) becomes empress, following the death of her cousin, Emperor Peter II.
March 12 – John Glas deposed from the Church of Scotland; the Glasite sect forms around him.
April 8 – Shearith Israel, the first synagogue in New York City, is dedicated.
May 15 – Charles Townshend, 2nd Viscount Townshend, retires from his role in the government of Great Britain, leaving Robert Walpole as sole and undisputed leader of the Cabinet (i.e., prime minister). In the new Walpole Ministry, Sir William Strickland, 4th Baronet, becomes Secretary at War, and Henry Pelham is Paymaster of the Forces. Spencer Compton, 1st Earl of Wilmington briefly becomes Lord Privy Seal.
May – At the urging of Sir William Gooch, the Virginia House of Burgesses passes the Tobacco Inspection Act of 1730 to regulate the quality of Virginian tobacco and establish inspection warehouses near plantations in the tidewater region.
Establishment of Wright's Ferry under the authority of the Province of Pennsylvania triggers Cresap's War – a nine-year-long conflict also known as the Maryland-Pennsylvania boundary dispute – the conflict mainly centered in Lancaster County, Pennsylvania and York County, Pennsylvania on either banks of the Susquehanna River.
July 8 – Earthquake affects Valparaiso in the Viceroyalty of Peru.
July 12 – Pope Clement XII succeeds Pope Benedict XIII as the 246th pope.
September 17 – Mahmud I (1730–1754) succeeds Ahmed III (1703–1730) as Ottoman Emperor.
October 22 – Construction of the Ladoga Canal linking the Neva and Svir Rivers, one of the first major navigable canals constructed in Russia, is completed.
March 16 – The Treaty of Vienna is signed between the Holy Roman Empire, Great Britain, the Dutch Republic and Spain.
April 2 – The town of Raynham, Massachusetts in Bristol County is entered as a new town by the governor and court of Massachusetts, New England, America.
April – British trader Robert Jenkins has his ear cut off by Spanish coast guards in Cuba, casus belli for the War of Jenkins' Ear in 1739.
July 1 – Benjamin Franklin and fellow-subscribers start the Library Company of Philadelphia.
Royal Colony of North Carolina Governor George Burrington asks the North Carolina General Assembly to pass an act establishing a town on the Cape Fear River, in what is seen as a political move to shift the power away from the powerful Cape Fear plantation class. The town is laid out in 1733 and incorporated as Wilmington in 1740.
English Captain Charles Gough rediscovers Gough Island in the South Atlantic.
Laura Bassi becomes the first official female university teacher on being appointed professor of anatomy at the University of Bologna at the age of 21.
John Bevis observes the Crab Nebula for the first time in the modern era.
The Royal Theatre of Mantua (Italy) is built by Ferdinando Galli Bibiena.
January 21st - Russia and Persia sign the Treaty of Riascha at Resht. Based on the terms of the agreement, Russia would no longer establish claims over Persian territories.
June 9 – James Oglethorpe is granted a royal charter for the colony of Georgia.
August – Mikhail Gvozdev in the Sviatoi Gavriil makes the first known crossing of the Bering Strait, from Cape Dezhnev to Cape Prince of Wales in Alaska.
September 16 – The magnitude 5.8 Montreal earthquake occurs in Montreal, Quebec, Canada.
December 7 – The original Theatre Royal, Covent Garden, London (the modern-day Royal Opera House) is opened.
December – 139 members of the Parlement of Paris exiled by order of King Louis XV secure their recall.
The Republic of Genoa regains Corsica.
The world's first lightship is moored at the Nore in the Thames Estuary of England.
This year's General Assembly of the Church of Scotland gives rise to the First Secession of 1733.
Benjamin Franklin, writing under the name Richard Saunders, begins publication of Poor Richard's Almanack. The annual publication will continue until 1758.
February 12 – British colonist James Oglethorpe founds Savannah, Georgia.
March – Molasses Act passed by Parliament of Great Britain, which aided in the negative opinions of the British by American Colonists.
April – Royal Colony of North Carolina Commissioners John Watson, Joshua Grainger, Michael Higgins and James Wimble plan the town of New Carthage (which would eventually become Wilmington, North Carolina on the east side of the Cape Fear River).
May 1 – Canton system is first introduced in Prussia.
May 29 – The right of Canadians to keep Indian slaves is upheld at Quebec.
July 30 – The first Freemasons lodge opens in what will become the United States of America.
October 5 – The election of Augustus III to succeed his father as King of Poland sparks the War of the Polish Succession.
November 23 – 1733 slave insurrection on St. John begins: Slaves from Akwamu rebel against owners in the Danish West Indies.
January 8 – Salzburgers, Lutherans who were expelled by the Roman Catholic Bishop of Salzburg, Austria, in October 1731, set sail for the British Colony of Georgia, in America.
March 12 – Salzburgers arrive at the mouth of the Savannah River, in the British Colony of Georgia.
June 17 – French troops take Philippsburg, but the Duke of Berwick is killed.
June 21 – In Montreal, New France, a black slave known by the French name of Marie-Joseph Angélique is tortured then hanged by the French authorities for allegedly setting a fire that destroyed part of the city.
June 30 – War of the Polish Succession: Russian troops take Gdańsk (German: Danzig), which had been besieged since February 1734. Gdańsk is captured after the failure of a French expedition to relieve the city.
November 5 – The Dzików Confederation is created in Poland.
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
January 23 – The Civil Code of 1734 is passed in Sweden.
January 26 – Stanislaus I of Poland abdicates his throne.
February 12 – Francis I, Holy Roman Emperor marries Maria Theresa of Austria, ruler of the Habsburg Empire.
April – The Genbun era begins in Japan. The era of Kyōhō Reforms ends.
March 8 – Nader Shah, founder of the Afsharid dynasty, is crowned Shah of Iran.
April 14 – The Porteous Riots erupt in Edinburgh after the execution of smuggler Andrew Wilson, when town guard Captain John Porteous orders his men to fire at the crowd. Porteous is arrested later.
May 8 – Frederick, Prince of Wales, marries Princess Augusta of Saxe-Gotha.
May 26 – Battle of Ackia: British and Chickasaw Native Americans defeat French troops.
June 8 – Leonhard Euler writes to James Stirling describing the Euler–Maclaurin formula, providing a connection between integrals and calculus.
June 19 – French Academy of Sciences expedition led by Pierre Louis Maupertuis, with Anders Celsius, begins work on measuring a meridian arc in Meänmaa of Finland.
July – Russo-Turkish War (1735–39): Russian forces under Peter Lacy storm the Ottoman fortress of Azov.
September 7 – An Edinburgh crowd drags John Porteous out of his cell in Tolbooth Prison and lynches him.
December 7 – Benjamin Franklin builds the first volunteer fire company in Philadelphia.
December 26 - Andrew Michael Ramsay gave an oration in which he related the heritage and internationalism of Freemasonry to that of the Crusades.
Neustrelitz becomes the capital of Mecklenburg-Strelitz.
Bushehr is founded in Persia.
The Belgrade Fortress is completed.
George Hamilton, 1st Earl of Orkney, becomes the first Field Marshal of Great Britain.
A fire in the Russian city of Saint Petersburg burns 2000 houses.
Fifty-three houses in the English town of Stony Stratford are consumed by fire.
One of the earliest records of use of a bathing machine is made at Scarborough in England.
Charles Marie de La Condamine, with François Fresneau Gataudière, makes the first scientific observations of rubber, in Ecuador.
Leonhard Euler produces the first published proof of Fermat's "little theorem".
Sir Isaac Newton's Method of Fluxions (1671), describing his method of differential calculus, is first published (posthumously) and Thomas Bayes publishes a defense of its logical foundations (anonymously).
Muhammad ibn Abd al-Wahhab writes the Kitab at-tawhidt, marking the beginning of Wahhabism.
The Haidamakas raid the shtetl of Pavoloch, killing 35.
May 28 – The planet Venus passes in front of Mercury. The event is witnessed during the evening hours by the amateur astronomer John Bevis at the Royal Greenwich Observatory. As of 2006, it is still the only such planet/planet occultation that has been directly observed.
June 21 – In Britain the Theatrical Licensing Act requires plays to be submitted to the Lord Chamberlain for censorship.
June 30 – Russo-Turkish War, 1735-1739: Russian forces under Field Marshal Munnich storm the Ottoman fortress of Ochakov and take prisoner 4,000 Turks.
July – Austria enters the Russo-Turkish War.
September 1 – The oldest existing English language newspaper in the world, The News Letter, is founded in Belfast, Ireland.
September 20 – Runner Edward Marshall completes his journey in the Walking Purchase, forcing the cession of 1,200,000 acres (4,900 km2) of Lenape-Delaware tribal land to the Pennsylvania Colony.
October – The first national stage in Sweden opens when the play Den svenska sprätthöken is performed in the native language, by the first native actors, on the stage of Bollhuset in Stockholm.
October 7 – A tropical cyclone strikes Bengal, India killing approximately 300,000.
October 16 – An earthquake with an estimated magnitude of 9.3 occurs off the shore of Russia's Kamchatka Peninsula. Tsunamis up to 60m (200 ft) high followed in the Pacific ocean.
November 4 – The Teatro di San Carlo, the oldest working opera house in Europe, is inaugurated in Naples, Italy.
Benjamin Franklin creates the Philadelphia Police Force – the first city-paid force.
The Georg August University of Göttingen is founded.
The direct male line of the Medici family becomes extinct with the death of Gian Gastone de' Medici, Grand Duke of Tuscany.
Richmond, Virginia is founded.
Our Lady of Guadalupe is designated the patron saint of Mexico City.
Lancaster County Prison, Lancaster, Pennsylvania is first constructed in response to the seven violent years preceding of the ongoing Cresap's War in the Maryland-Pennsylvania boundary dispute and war.
February 4 – Court Jew Joseph Süß Oppenheimer is executed in Württemberg.
March/April – Emperor Iyasu II of Ethiopia is defeated by the Funj at the Battle of the Dindar River.
April 15 – Serse, an Italian opera by George Frideric Handel, premieres in London.
May 24 – John Wesley experiences a spiritual rebirth at a Moravian Church meeting in Aldersgate in the City of London, essentially launching the Methodist movement; the day is celebrated annually by Methodists as Aldersgate Day. His younger brother Charles had a similar experience three days earlier.
September 18 – Samuel Johnson composes his first solemn prayer (published 1785).
November 18 – The Treaty of Vienna is ratified, ending the War of the Polish Succession. Under the terms of the treaty, Stanisław Leszczyński receives Lorraine in exchange for renouncing the Polish throne.
Specific date unknown: Carl Philipp Emanuel Bach - having completed a law degree - was hired as a court musician by Crown Prince Frederick of Prussia, the future Frederick the Great. Bach was to remain in Frederick's service until 1768.
Russo–Turkish War (1735–1739): The Russian army fails to cross the Dniester. They are decimated by plague.
China's Qing government announces that all western businessmen have to use the Cohong in Guangzhou to trade.
The excavation of Herculaneum, a Roman city buried by Vesuvius in AD 79, begins.
Pierre Louis Maupertuis publishes Sur la figure de la terre, which confirms Newton's view that the earth is an oblate spheroid slightly flattened at the poles.
Jacques de Vaucanson presents the world's first automaton, The Flute Player (1737) to the French Academy of Sciences.
Black Forest clockmaker Franz Ketterer produces one of the earliest cuckoo clocks.
Holy Royal Arch is founded.
Rémy Martin is granted exclusive permission by King Louis XV to plant new vineyards for impressing him with the quality of his cognac.
January 1 – Bouvet Island is discovered by French explorer Jean-Baptiste Charles Bouvet de Lozier in the South Atlantic Ocean.
February 24 – Battle of Karnal: The army of Iranian ruler Nader Shah defeats the forces of the Mughal emperor of India, Muhammad Shah.
March 20 – Nader Shah occupies Delhi in India and sacks the city, stealing the jewels of the Peacock Throne, including the Koh-i-Noor.
May 12 – John Wesley lays the foundation stone of the New Room, Bristol in England, the world's first Methodist meeting house.
June 2 – The Royal Swedish Academy of Sciences is founded in Stockholm, Sweden.
September 9 – The Stono Rebellion, a slave rebellion, erupts near Charleston, South Carolina.
September 18 – The Treaty of Belgrade brings the Austro-Russian–Turkish War (1735–39) to an end.
October 3 – The Treaty of Niš is signed.
October 17 – The Foundling Hospital is created in London by royal charter.
October 23 – War of Jenkins' Ear: Great Britain declares war on Spain.
November 20–22 – War of Jenkins' Ear: Battle of Porto Bello: British marine forces capture the Panamanian silver exporting town of Portobelo from the Spanish.
Ecuador becomes a part of New Granada.
84,000 farmers revolt in the province of Iwaki in Japan.
Plinian eruption of Mount Tarumae volcano in Japan.
The first Bible in Estonian is published.
February 22, 1732 – George Washington, 1st President of the United States (d. 1799)