4. South Georgia and the South Sandwich Islands
This small group of islands has been the source of much animosity between Argentina and the UK. The UK has claimed sovereignty over South Georgia since 1775 when James Cook sailed there. Then in 1908 the UK annexed both South Georgia and the South Sandwich Islands. However, an Argentinian whaling company set up their operations in South Georgia in 1904.
Under British control South Georgia and the South Sandwich Islands became part of the Falklands. In 1906, the Argentinian whaling company signed a lease and began using British whaling licenses. Despite the whaling company changing their flag to that of the British, the Argentinian government made a claim on South Georgia in 1927 and on the rest of the islands in 1938. The UK, in an attempt to bring peace and settle the matter, have offered to take it up with the International Court of Justice 4 times between 1945 and 1955. Argentina refused. Argentina actually maintained a summer station and a naval base on Thule Island. The British discovered the base in 1974 and protested the issue, but the UK never forced them to leave until 1982 after the Falklands war.
On March 19th, 1982 a group of Argentinians posed as scrap merchants and occupied an abandoned whaling station. Then on April 3 they moved in on Grytviken and took over the city. The occupation did not last long as British forces took back the city on April 25th. After that the British removed the Argentinians from Thule by force.
In 1985, South Georgia and the South Sandwich Islands broke away from the Falkland Islands and became a British overseas territory. The issue was brought up again in 2010 when the President of Venezuela, Hugo Chavez, insisted that Queen Elizabeth II return the islands to Argentina. The British government refused and Argentina still maintains its claim over South Georgia and the South Sandwich Islands.