How "special" are closed subvarieties of abelian varieties over number fields? (Dimension 1 is easy.)

For example: Are there interesting families of varieties of general type which are not closed subvarieties of abelian varieties?

This is motivated by Faltings' articles *Diophantine approximation on abelian varieties.* Ann. of Math. (2) 133 (1991), no. 3, 549–576 and *The general case of S. Lang's conjecture.* Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), 175–182, Perspect. Math., 15, Academic Press, San Diego, CA, 1994.: To which varieties does Faltings' theorem apply?

(There is the related question Properties of subvarieties of a simple abelian variety.)