  We study smooth complex projective varieties $X$ of maximal Albanese
dimension and of general type satisfying with vanishing holomorphic Euler
characteristic. We prove that the Albanese variety of $X$ has at least three
simple factors. Examples were constructed by Ein and Lazarsfeld, and we prove
that in dimension 3, these examples are (up to abelian \'etale covers) the only
ones. By results of Ueno, another source of examples is provided by varieties
$X$ of maximal Albanese dimension and of general type with $p_g(X)=1$. Examples
were constructed by Chen and Hacon, and again, we prove that in dimension 3,
these examples are (up to abelian \'etale covers) the only ones. We also
formulate a conjecture on the general structure of these varieties in all
dimensions.
