Classically, complex projective curves (also called "Riemann surfaces") fall into three distinct classes: rational, elliptic and hyperbolic. This geometric trichotomy can be defined in various ways, topological or analytical. But it can be defined also arithmetically, when the curve is defined over a number field k. This trichotomy then actually reduces to a dichotomy: "special" (rational and elliptic curves: those having a dense set of k-rational points if k is large enough) versus "hyperbolic" (those for which this set remains finite for any k, by Mordell's conjecture, proved by G. Faltings).
This trichotomy, as well as the above more fundamental dichotomy, can be defined in purely geometrical terms for complex projective manifolds X of higher dimensions as well. Special and hyperbolic manifolds enjoy quite opposite properties. Conjecturally, the special manifolds should be, just as in the case of curves, exactly the ones with a dense set of k-rational points. While, according to a conjecture of S. Lang, the "hyperbolic" manifolds should be exactly the ones having a finite set of k-rational points (outside some fixed strict algebraic subset).
An arbitrary variety X can be canonically and functorially decomposed by a single fibration, into its special (the fibres) and hyperbolic (the orbifold base) parts. This decomposition naturally leads to a conjectural qualitative description of the distribution of k-rational points for any variety X.