The Tiwi Islands are composed of Melville Island and Bathurst Island, with a combined area of 8,320 sq.km. They have a combined population of just over 2000 people, over 90% of whom are Aboriginal. The largest settlement, Nguiu (pron. "new-you") on Bathurst, has a population of 1,500.
The Catholic mission established on the Tiwi Islands in 1911 greatly influenced the culture of the Tiwi Islanders. The Islanders also played a role in World War II, capturing Japanese pilots who were shot down during bombing raids on Darwin. Many Tiwi Islanders are prolific artists who produce distinctive art, pottery, sculptures and wooden carvings.
Visiting the Tiwi Islands requires a permit from the Tiwi Islands Land Council . For casual visitors, the only way to arrange one is through one of two designated tour operators:
Once your permits are sorted out, there are two ways to reach the Tiwis from Darwin: 20-minute flights on Fly Tiwi  (several times daily, from $120/one-way), or a two-hour catamaran journey on the Arafura Pearl  (thrice weekly, from $75/one-way).
The biggest event of the year on the famously football-crazy (Australian rules, of course) islands is the Tiwi Islands Football League Grand Final in late March, which draws crowds above 3000 — more than the islands' combined population! While permits were previously not required to visit at this time, this is apparently no longer the case, and in any case you'll need to plan well in advance to secure your seats. The Tiwi Islands Art Sale is held at the same time.
The only way out is back to Darwin.