- Jun 4 Cyprus ceded by Turkey to Britain for administrative purposes
- Jun 5 American outlaw John Wesley Hardin convicted of the murder of Deputy Sheriff Charles Webb, sentenced to 25 years in prison
- Jun 8 12th Belmont: L Hughes aboard Duke of Magenta wins in 2:43.5
- Jun 11 DC is given a new government by Congress, 3 commissioners appointed by president (change in 1974)
- Jun 13 Congress of Berlin begins, determines the territories of the states in the Balkan peninsula following the Russo-Turkish War of 1877–78
First Moving Images
Jun 15 World's first moving pictures caught on camera (used 12 cameras, each taking 1 picture) done to see if all 4 of a horse's hooves leave the ground
- Jun 22 Swedish ship SS Vega with explorer Nils Adolf Erik Nordenskiöld sets out to find the Northeast Passage (succeeds 1879)