For a curve defined over a finite field, one can associate to it a polynomial with integer coefficients. This Frobenius polynomial has many remarkable properties and encodes much of the curve's arithmetic information. For a curve defined over the rationals, one obtains a polynomial for almost every prime p by reducing modulo p. These polynomials have several interesting statistical distributions as we vary the prime; one of them, the Sato-Tate conjecture, we will explain.