Genus one curves, defined over the rationals, need not have rational points. The set of all such curves, whose Jacobian is a fixed elliptic curve E, form a group, called the Weil-Chatelet group. It has an important subgroup, the Tate-Shafarevich group, formed by those curves which have points over all completions of the rationals. This talk will address two aspects of the arithmetic of genus one curves: (1) (with J. Stix) the divisibility of the Tate-Shafarevich group inside the Weil-Chatelet group; (2) (with A. Wiles) work in progress on the existence of points defined over number fields with solvable Galois group over the rationals. Earlier work, also with Wiles, proved existence when the curve represents an element of the Tate-Shafarevich group; we now aim to extend this to the whole Weil-Chatelet group.