- Jun 5 The Battle of New Ross: The attempt to spread United Irish Rebellion into Munster is defeated.
- Jun 7 Jews of Pesaro Italy fast commemorating murder of Jews
- Jun 7 Thomas Malthus publishes the first edition of his influential 'Essay on the Principle of Population' (date of the unsigned preface)
- Jun 13 Mission San Luis Rey de Francia founded in California
- Jun 25 US passes Alien Act allowing president to deport dangerous aliens