the goal of the talk is to explain some geometric results on quasi-projective manifolds from the perspective of carathéodory metrics and distances.  we will study some conjectures of lang on manifolds which satisfied some carath\'eodory conditions. the results are also used to study hyperbolicity of suitable compactifications of the non-compact manifolds involved.  as applications, we also prove some statements to the effect of non-existence of level structures on manifolds such as abelian varieties over function fields, as well as the  so-called volume estimates for mapping of curves into the manifolds involved. most of the results to be presented are joint work with kwok-kin wong.