Which almost complex manifolds admit a complex structure? I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since Yau's list was published, so what is the status of this problem today?
Obviously it isn't hasn't been shown to be true, because we're still looking for complex structures on the six-sphere, but I have a vague feeling of having read that this doesn't hold. So do we know any counterexamples to this question? If not, then is anyone working on this problem?
Also, Yau only stated the problem for manifolds of dimension $n \geq 3$. We know this is true in dimension one, because there we have isothermal coordinates which give complex structures, but why didn't Yau mention almost complex surfaces? Do we know this holds there, or are there counterexamples in dimension 2?
 A: There are actually counterexamples in real dimension $4$.
The first examples of compact almost complex $4$-manifolds admitting no complex structure were produced by Van de Ven in his paper "On the Chern numbers of some complex and almost-complex manifolds". 
In fact, he obtained restrictions on the Chern numbers of an algebraic surface and constructed some almost complex $4$-manifolds violating them, hence showing that no almost complex structure in these examples could be integrable.    
Later on, Brotherton constructed some counterexamples with trivial tangent bundle, see the article "Some parallelizable 4-manifolds not admitting a complex structure". 
A: In complex dimension 3 or more it is still an open conjecture
(which was re-stated Yau a couple of years ago in his UCLA lectures).
There is not a single known example of an almost complex manifold
of dimension $\geq$ 3 not admitting a complex structure.
In dimension 2 it is easy, of course, because 
the non-Kähler complex surfaces are understood
much better than Kähler ones: every non-Kähler
surface with $b_1 >1$ is diffeomorphic to a blow-up of a 
locally trivial elliptic fibration over a curve. Hence any
4-dimensional compact almost complex manifold 
with odd $b_1 >1$ and a fundamental group not virtually 
isomorphic (*) to a cross-product of a fundamental group of a curve
and $\mathbb{Z}$, cannot be a complex surface.
(*)  Here "virtually isomorphic" means "isomorphic up to a 
finite index subgroup".
