In this talk I will introduce elliptic curves and discuss recent work with Barry Mazur on ranks of elliptic curves in families of quadratic twists. One application of our results, using ideas of Poonen and Shlapentokh, is that if a standard conjecture about elliptic curves (the Shafarevich-Tate conjecture) holds then Hilbert's 10th problem for the ring of integers of any number field has a negative answer. (Hilbert's 10th problem for a ring R asks whether there is an algorithm to decide whether a multivariate polynomial with coefficients in R has solutions in R. Around 1970 Matijasevich, building on work of Davis, Putnam, and Robinson, proved that no such algorithm exists when R is the ring of rational integers. This answered Hilbert's original question, but for many other rings of interest the question is still open.)