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We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields 
$k$ of characteristic zero. For example, given a ramified cover 
$\pi : X \to A$, where 
$A$ is an abelian variety over 
$k$ with a dense set of 
$k$-rational points, we prove that there is a finite-index coset 
$C \subset A(k)$ such that 
$\pi (X(k))$ is disjoint from 
$C$. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the inverse Galois problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.
