Are there many almost complex structures on a (complex) manifold?

Question

I guess one can have many almost complex structures on a manifold, can someone give me an example? How about when the manifold is complex? is the almost complex structure induced by the complex structure the only one?

Answer 1

One important example is the 1-dimensional one: For Riemann surfaces, all almost complex structures are integrable (see e.g. Theorem 11.1.6 of these notes). On the other hand, surfaces generally admit many different complex structures, and these are parametrized by Teichmüller space.