April 6 – The rebellion by Túpac Amaru II, against the Spanish colonial government of Peru, is ended as Tupac, his wife and two of his sons are captured at Checacupe. 
April 10— Future U.S. President Andrew Jackson, age 14, is slashed by a British officer's sword at his home near Waxhaw, North Carolina, after refusing to clean the officer's boots, an event that leaves physical and psychological scars. 
April 18— Future New York mayor James Duane, North Carolina representative William Sharpe and future Connecticut governor Oliver Wolcott deliver the first report to the U.S. Continental Congress about the national debt and report it to be 24,057,157 and 2/5 dollars. 
May 18 – A Spanish army sent from Lima puts down the Inca rebellions, and captures and savagely executes Túpac Amaru II.
June 4 – The commission[which?] agrees to the rebels'[where?] terms: reduction of the alcabala and of the Indians' forced tribute, abolition of the new taxes on tobacco, and preference for Criollos over peninsulares in government positions.
January 23 – The Laird of Johnstone (George Ludovic Houston) invites people to buy marked plots of land which, when built upon, form the planned town of Johnstone, Scotland, to provide employment for his thread and cotton mills.
July 16–August 29 – The Masonic Congress of Wilhelmsbad, Germany, one of history's most important ever secret society congresses, takes place. High-degree Freemasons from the whole of Europe spend the time deliberating the fate of the rite of Strict Observance, and hierarchy of the governing bodies of world Freemasonry, at the Hanau-Wilhelmsbad spa.
October 10— Welsh actress Sarah Siddons, the pre-eminent star of the English stage, makes a triumphant return to the theatre in the title role of David Garrick's new play, Isabella, or The Fatal Marriage. 
October 18— The first franking privilege is granted for official correspondence to be sent at no charge to and from members of the Confederation Congress, at government expense, during periods when the Congress is in session. 
In China, the Siku Quanshu is completed, the largest literary compilation in China's history (surpassing the Yongle Encyclopedia of the 15th century). The books are bound in 36,381 volumes (册) with more than 79,000 chapters (卷), comprising about 2.3 million pages, and approximately 800 million Chinese characters.
The volcanoLaki in Iceland begins an 8-month eruption, starting the chain of natural disasters known as the Móðuharðindin, killing tens of thousands throughout Europe, including up to 33% of Iceland's population, and causing widespread famine. It has been described as one of "the greatest environmental catastrophes in European history".
General George Washington sends a letter to the 13 governors of the confederation of United States regarding the needs of the nation.
July 29 – The United States and the Kingdom of France sign a convention for establishing diplomatic relations and "determining the functions and prerogatives of their respective consuls, vice consuls, agents, and commissaries". 
September 19 – In France, the Robert brothers (Anne-Jean Robert and Nicolas-Louis Robert) and a Mr. Collin-Hullin (whose first name is lost to history) become the first people to fly more than 100 km or 100 miles in the air, lifting off from Paris and landing 6 hours and 40 minutes later near Bethune after a journey of 186 kilometres (116 mi).
May 20 – The Northwest Ordinance of 1785, setting the rules for dividing the U.S. Northwest Territory (later Ohio, Indiana, Illinois, Wisconsin and Michigan) into townships of 36 square miles apiece, is passed by the Confederation Congress. Walter G. Robillard and Lane J. Bouman, Clark on Surveying and Boundaries (LexisNexis, 1997) The survey system will later be applied to the continent west of the Mississippi River. 
June 15 – After several attempts, Jean-François Pilâtre de Rozier and his companion, Pierre Romain, set off in a balloon from Boulogne-sur-Mer, but the balloon suddenly deflates (without the envelope catching fire) and crashes near Wimereux in the Pas-de-Calais, killing both men, making it the first fatal aviation disaster.
October 17 – The Commonwealth of Virginia stops the importation of new African slaves by declaring that "No persons shall henceforth be slaves within this commonwealth, except such as were so on the seventeenth day of October, 1785, and the descendants of the females of them." 
November 23 – John Hancock of Massachusetts, the former President of the Continental Congress, is selected as the new President of the Congress of the Confederation, but is unable to take office because of illness. 
April 25 – The United States and the Kingdom of Portugal sign their first commercial treaty, but it is never ratified. 
April 27 – British astronomer William Herschel publishes his first list of his discoveries, Catalogue of One Thousand New Nebulae and Clusters of Stars; two additional books are published in 1789 and 1802. 
October 10 – The Confederation Congress of the United States directs backpay for seven months for Virginia officers who have been waiting since 1782. 
October 12 – King George III of the United Kingdom appoints Captain Arthur Phillip as the first Governor of New Holland, which comprises the area of modern Australia from the 135th meridian east to the east coast and all adjacent islands in the Pacific Ocean. 
October 16 – The Confederation Congress establishes the United States Mint to make common coinage and currency for the U.S., to replace individual state coins. 
March 28 – In the British House of Commons, Henry Beaufoy files the first motion to repeal the Test Act 1673, which restricts the rights of non-members of the Church of England. ; Beaufoy's motion is rejected, and the Act is not repealed until 1829.
April 13 – America's first recorded riot, the 'Doctors' Mob', begins. Residents of Manhattan are angry about grave robbers stealing bodies for doctors to dissect. The rioting is suppressed on April 15.
November 8 – Voting takes place in the 11 states that have ratified the United States Constitution for the first U.S. Senators; in Virginia, Richard Henry Lee and William Grayson, both anti-federalists, receive the highest number of votes in the Virginia Senate. 
November 20 – In the United Kingdom, the Houses of Parliament are given the first formal report by Prime Minister Pitt of the mental illness of King George III. Parliament adjourns for two weeks to await the results of examinations by royal physicians. 
November 25 – Fifty consecutive days of temperatures below freezing strike France, a record that would be unbroken more than 200 years later. 
April 6 – At Federal Hall, the United States Senate attains its first quorum, and elects John Langdon of Pennsylvania as its first President pro tempore. Later that day, the Senate and the House of Representatives meet in joint session for the first time, and the electoral votes of the first U.S. Presidential election are counted. General George Washington is certified as President-elect, and John Adams is certified as Vice-President elect.