A basic question in arithmetic statistics is:  what does the Selmer group of a random abelian variety look like?  This question is governed by the Poonen-Rains heuristics, later generalized by Bhargava-Kane-Lenstra-Poonen-Rains, which predict, for instance, that the mod p Selmer group of an elliptic curve has size p+1 on average.  Results towards these heuristics have been very partial but have nonetheless enabled major progress in studying the distribution of ranks of abelian varieties.