An almost complex structure on a smooth manifold is a smooth section such that, over each point , is a linear complex structure on that tangent space under the canonical identification . Equivalently this is a reduction of the structure group of the tangent bundle to the complex general linear group along .
A complex structure on a smooth manifold is the structure of a complex manifold on . Every such defines an almost complex structure and almost complex structures arising this way are called integrable .
The Newlander-Nierenberg theorem states that an almost complex structure on a smooth manifold is integrable precisely if its Nijenhuis tensor? vanishes, .
Every alomost complex structure canonically induces a spin^c-structure. See there for more.
A discussion of deformations of complex structures is in