Solving Diophantine equations is a central and ancient problem in mathematics. The easiest case, that of curves, is still very difficult. In this talk I'll give a survey about what is known for this “easiest” case, starting with conics and leading up to the recent work of Bhargava and his various coauthors on the rank conjecture for elliptic curves and the conjecture of Birch and Swinnerton-Dyer.