See Nurikabe for a different puzzle also sometimes called "Islands".
Shade in a polyomino in each thickly outlined region such that no two polymoinoes touch each other by a side, no two regions sharing an edge contain polyominoes of the same size, and a numbered region contains a polyomino of that size.
(Rules and example from WPC 2017 IB)
History of the puzzle
First appeared on Nikoli volume 117 (2006). Invented by 木兄 (Mokuani). Originally named 島国 (Shimaguni, lit. "Island country").