  We study the space of closed anti-invariant forms on an almost complex
manifold, possibly non compact. We construct families of (non integrable)
almost complex structures on $\R^4$, such that the space of closed
$J$-anti-invariant forms is infinite dimensional, and also $0$- or
$1$-dimensional. In the compact case, we construct $6$-dimensional almost
complex manifolds with arbitrary large anti-invariant cohomology and a
$2$-parameter family of almost complex structures on the Kodaira-Thurston
manifold whose anti-invariant cohomology group has maximum dimension.
