We will discuss joint work with Corvaja, Demeio, Lombardo, and Zannier in which we extend Hilbert's irreducibility theorem (for rational varieties) to the setting of abelian varieties. Roughly speaking, given an abelian variety $A$ over a number field $k$ and a ramified covering $X$ of $A$, we show that $X$ has "less" $k$-rational points than $A$.