A basic problem in number theory, going back to Diophantos, is to understand the rational solutions of systems of polynomial equations with integer coefficients. This is recast usually in terms of rational points on the algebraic variety $V$ defined by the polynomial system. An intriguing principle is that the geometry of the complex variety $V(C)$ has a strong bearing on the set $V(Q)$ of rational points. A conjecture of Lang predicts a strong finiteness result when $V(C)$ is hyperbolic, which we will explicate when $V(C)$ is uniformized by the unit complex ball, representing some joint work with M. Dimitrov.