Album Release 
- Jun 2 The Bhutan Broadcasting Service brings television transmissions to the Kingdom for the first time
- Jun 4 NHL Western Conference Final: Dallas Stars beat Colorado Avalanche, 4 games to 3
- Jun 5 131st Belmont: Jose Santos aboard Lemon Drop Kid wins in 2:27.88
- Jun 6 In Australian Rules Football, Tony Lockett breaks the record for career goals, previously 1299 by Gordon Coventry and which had stood since 1937.
- Jun 6 Largest jailbreak in Brazilian history at the Putim maximum security prison in Brazil, 345 prisoners run from the main gate. In the ensuing manhunt, two fugitives are killed and five innocent bystanders are accidentally jailed.
- Jun 6 53rd Tony Awards: "Fosse" and "Side Man" win
Tennis Open
Jun 6 98th Men's French Open: Andre Agassi beats Andrei Medvedev (1-6, 2-6, 6-4, 6-3, 6-4)
Andre Agassi
- Jun 6 98th Women's French Open: Steffi Graf beats Martina Hingis (4-6, 7-5, 6-2)
- Jun 9 Kosovo War: the Federal Republic of Yugoslavia and North Atlantic Treaty Organization sign a peace treaty.
Agreement of Interest
Jun 10 Kosovo War: NATO suspends its air strikes after Slobodan Milošević agrees to withdraw Serbian forces from Kosovo.
Slobodan Milosevic
Event of Interest
Jun 19 At about 4:30 pm, Stephen King is hit by a car on the shoulder of Route 5, in Lovell, Maine. He would suffer numerous injuries, including a collapsed right lung, multiple fractures of his right leg, scalp laceration and a broken hip
Stephen King
Event of Interest
Jun 19 Stanley Cup: Dallas Stars defeat Buffalo Sabres 4-2, Brett Hull clinches series with controversial "No Goal"
Brett Hull
Golf Major
Jun 20 99th US Golf Open: Payne Stewart shoots a 279 at Pinehurst CC
Payne Stewart
- Jun 22 Former Mpumalanga premier, Ndaweni Mahlangu causes a storm within political circles with his now infamous statement, "It is acceptable for politicians to lie", South Africa
- Jun 25 53rd NBA Championship: San Antonio Spurs beat New York Knicks, 4 games to 1
- Jun 25 In his first state of the nation address, South African President Thabo Mbeki promises to tackle rampaging crime; the nation's murder rate is the third highest in the world and more than 49,000 cases of rape were reported in 1998
- Jun 27 ATB go to No.1 on the UK singles chart with '9pm, Till I Come'
- Jun 27 The Chemical Brothers go #1 on the UK album chart with 'Surrender', their second #1 album
- Jun 27 45th LPGA Championship won by Juli Inkster