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Let π be a permutation of the set rns " t1, 2, . . . , nu. Two disjoint orderisomorphic subsequences of π are called twins. How long twins are contained in every permutation? The well known Erdős-Szekeres theorem implies that there is always a pair of twins of length Ωp ? nq. On the other hand, by a simple probabilistic argument Gawron proved that for every n ě 1 there exist permutations with all twins having length Opn 2{3 q.He conjectured that the latter bound is the correct size of the longest twins guaranteed in every permutation. We support this conjecture by showing that almost all permutations contain twins of length Ωpn 2{3 q. This was also proved recently by Bukh and Rudenko (see Remark 1.2 below on the interrelation between these two results).We also study several variants of the problem with diverse restrictions imposed on the twins. For instance, if we restrict attention to twins avoiding a fixed permutation τ , then the corresponding extremal function equals Θp ? nq, provided that τ is not monotone. In case of block twins (each twin occupies a segment) we prove that it is p1 `op1qq log n log log n , while for random permutations it is twice as large. For twins that jointly occupy a segment (tight twins), we prove that for every n there are permutations avoiding them on all segments of length greater than 24.