On group structures realized by elliptic curves over arbitrary finite fields

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection which correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory, some are conditional under certain widely believed conjectures, and others are purely heuristic in nature.