The British Overseas Territories are the remnants of the British Empire. Its comprised of Anguilla, Bermuda, the British Virgin Islands, the Cayman Islands, the Falkland Islands, Gibraltar, Montserrat, the Pitcairn Islands, St Helena (and its dependencies Ascension and Tristan da Cunha), and the Turks and Caicos Islands. It also includes the following: the British Antarctic Territory, the British Indian Ocean Territory, South Georgia and the South Sandwich Islands; and the Sovereign Base Areas of Akrotiri and Dhekelia in Cyprus.
These countries have a special relationship with the United Kingdom in that, even though most of these territories have their own governments, the British Government is responsible for defense and external affairs. With the exception of Monserrat and St. Helena (both due to unfortunate circumstances)1, none of the territories receive budgetary assistance. As a matter of fact, some of them (Bermuda and Cayman Islands for instance) enjoy higher GDPs2 per capita than the United Kingdom.
Why not grant independence to these territories?
The British Overseas Territories have chosen to remain British. In most cases, these territories enjoy a win-win situation by being linked to Britain. They have most of the benefits of independence in that they have responsibility for their own econmonies and internal affairs.