On a smooth complex projective variety of $\dim X=n$, we have $n$ complex tori associated to it via $J^k(X)=F^kH^{2k-1}(X,\mathbb{C})/H_k(X,\mathbb{Z})$ (assuming I've got all the indices right) called the $k$th intermediate Jacobian.

If $k=1$, we have $J^1(X)=H^{1,0}/H_1$, and so $J^1(X)\cong Jac(X)$ is an abelian variety (the bilinear form is a polarization because it has to be definite on each piece of the Hodge decomposition (I think) ) and is in fact isomorphic as PPAV's to the Jacobian of the variety.

If $k=n$, we have $H^{2n-1,1}/H_{2n-1}$, which is also a PPAV, and is in fact the Albanese of $X$.

The ones in the middle, however, the "true" intermediate Jacobians, are generally only complex tori. One example of an application is that Clemens and Griffiths proved that cubic threefolds are unirational but not rational using $J^2(X)$ for $X$ a cubic threefold.

So, what information do the intermediate Jacobians contain? I've been told that we don't really know much about that, but what is known, beyond Clemens/Griffiths?