  We give a sharp lower bound for the selfintersection of a nef line bundle $L$
on an irregular variety $X$ in terms of its continuous global sections and the
Albanese dimension of $X$, which we call the Generalized Clifford-Severi
inequality. We also extend the result to nef vector bundles and give a slope
inequality for fibred irregular varieties. As a byproduct we obtain a lower
bound for the volume of irregular varieties; when $X$ is of maximal Albanese
dimension the bound is $Vol(X) \geq 2 n! \chi\omega_X$ and it is sharp.
