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?