In joint work with Barry Mazur, we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields.  We give sufficient conditions for an elliptic curve to have twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) with a given 2-Selmer rank.  As a consequence, under appropriate hypotheses there are many twists with Mordell-Weil rank zero, and (assuming the Shafarevich-Tate conjecture) many others with Mordell-Weil rank one.  Another application of our methods, using ideas of Poonen and Shlapentokh, is that if the Shafarevich-Tate conjecture holds then Hilbert's 10th problem has a negative answer over the ring of integers of any number field.