I will present recent joint work with N. Chen about dominant rational maps from products of curves to surfaces with p_g=q=0. The gonality of an algebraic curve C is the minimal degree of a non-constant morphism from C to the projective line. Our main result is that under some assumptions the minimal degree of a dominant rational map from a product of two curves to a surface with p_g=q=0 is the product of their gonalities. In particular, a product of hyperelliptic curves of general type does not admit dominant rational maps of degree less than 4 to P^2. I will finish by presenting open problems and some strategies to attack them.