Relations with Argentina
Main article: Sovereignty of the Falkland Islands

The dispute over control of the islands has continued since the war. Diplomatic relations between Argentina and the UK were resumed in 1992, and embassies were reopened in London and Buenos Aires. In 1994, Argentina added its claim to the islands to the Argentine constitution, stating that this claim must be pursued in a manner "respectful of the way of life of their inhabitants and according to the principles of international law"(see: 1994 reform of the Argentine Constitution).

Falkland Islanders were granted full British citizenship from 1 January 1983 under the British Nationality (Falkland Islands) Act 1983. As Argentina considers the Falklands to be Argentine territory, they also consider the Falkland Islanders to be Argentine citizens through the system of jus soli operated under Argentine nationality law, though this is rejected by the islanders themselves.

In 1998, in retaliation for the arrest in London of the former Chilean president Augusto Pinochet, the Chilean government banned flights between Punta Arenas and Port Stanley, thus isolating the islands from the rest of the world. Uruguay and Brazil refused to authorise direct flights between their territories and Port Stanley, forcing the Islands' government to enter negotiations with the Argentine government which led to Argentina authorising direct flights between its territory and Stanley, on condition that Argentine citizens be allowed on the islands. One flight a month, operated by LAN Airlines, travels between RAF Mount Pleasant on East Falkland and Río Gallegos in Santa Cruz Province, Argentina.

Since the war, successive Argentine governments have stated their intention to pursue their claim to the islands by peaceful means. On the 22nd anniversary of the war, Argentina's President Néstor Kirchner gave a speech insisting that the islands would once again be part of Argentina. Kirchner, campaigning for president in 2003, regarded the islands as a top priority. In June 2003 the issue was brought before a United Nations committee, and attempts have been made to open talks with the United Kingdom to resolve the issue of the islands.

In 2007 (exactly 25 years after the Argentine invasion), Argentina renewed its claim over the Falkland Islands, asking for the UK to resume talks on sovereignty. In March 2009, British Prime Minister Gordon Brown stated in a meeting with Argentine President Cristina Fernández that there would be no talks over the future sovereignty of the Falkland Islands. As far as the governments of the UK and of the Falkland Islands are concerned, there is no issue to resolve. The Falkland Islanders themselves are almost entirely British and maintain their allegiance to the United Kingdom.

On 22 September 2007, The Guardian reported the UK government was preparing to stake new claims on the sea floor around the Falklands and other UK remote island possessions, in order to exploit natural resources that may be present. In October 2007, a British spokeswoman confirmed that Britain intended to submit a claim to the UN to extend seabed territory around the Falklands and South Georgia, in advance of the expiry of the deadline for territorial claims following Britain's ratification of the 1982 Law of the Sea Convention. If the claim is disputed, the UN will suspend the claim until the dispute is settled. The claim is largely theoretical and does not affect the Antarctic Treaty or confer new rights upon Britain. Neither does it permit the exploitation of oil or gas reserves, since these are banned by a protocol to the treaty. It would enable Britain to police fishing within the zone to prevent over exploitation of natural resources by commercial fishing in line with Britain's obligations under the treaty. Nevertheless many commentators have criticised the move for going against the spirit of the Antarctic Treaty.[38] Argentina has indicated it will challenge any British claim to Antarctic territory and the area around the Falkland Islands and South Georgia. Argentina made a similar claim in 2009, and the United Kingdom quickly protested these claims.