- Jun 1 Glorious First of June; first naval battle between Britain (under Admiral Lord Howe) and France (Vice-Admiral Louis Thomas Villaret de Joyeuse) during French Revolutionary Wars. Britain gains tactical win.
- Jun 5 US Congress passes the Neutrality Act, banning Americans from serving in foreign armed forces
- Jun 10 Bethel African Methodist Episcopal Church in Philadelphia forms
- Jun 10 France revolutionary regime begins trials
- Jun 16 1st stone laid at biggest Dutch grain windmill De Walvisch in Schiedam
Jun 26 Battle of Fleurus: major victory by forces of the First French Republic under General Jean-Baptiste Jourdan over the Coalition Army (Great Britain, Hanover, Dutch Republic, and Habsburgs) first use of reconnaissance balloon
- Jun 30 Seige of Fort Recovery, Ohio, Anglo American army holds off an attack on the fort by an Indian confederacy