We define several variants of the notion of twins (and multiple twins) in permutations and study their presence in both, permutations and random permutations. The main question of how large twins can be found in these structures has gotten so far rather imprecise answers with lower and upper bounds rarely matching each other. Consequently, during the lecture several open problems will be stated.
This is joint work with A. Dudek and J. Grytczuk.