Why do some polynomial equations have only finitely many solutions in the integers? Lang-Vojta's conjecture provides a conjectural answer and relates this number-theoretic question to complex geometry. Indeed, conjecturally, a variety has only finitely many rational points if and only if it is hyperbolic. I will start out this talk explaining the Lang-Vojta conjectures, and will then present new results on dynamical systems of hyperbolic varieties, rational points on ramified covers of abelian varieties, the fundamental group of a variety covered by many pointed curves, and rigidity results for families of canonically polarized varieties. These results are mathematically independent, but all guided by the conjectures of Lang-Vojta.