  We study a special type of almost complex structures, called pure and full
and introduced by T.J. Li and W. Zhang, in relation to symplectic structures
and Hard Lefschetz condition. We provide sufficient conditions to the existence
of the above type of almost complex structures on compact quotients of Lie
groups by discrete subgroups. We obtain families of pure and full almost
complex structures on compact nilmanifolds and solvmanifolds. Some of these
families are parametrized by real 2-forms which are anti-invariant with respect
to the almost complex structures.
