Abstract: Symplectic manifolds can be naturally provided with compatible almost-complex structures, which allow one to define pseudo-holomorphic curves. On the one hand, they are a powerful tool in probing symplectic geometry and one of the ingredients for defining Floer cohomology. On the other hand, on the complex side, they lead to the definition of (almost)-complex hyperbolicity. Going further, Floer cohomology allows one to define a notion of symplectic hyperbolicity. This naturally raises the issue of possible links between the symplectic hyperbolicity of a symplectic manifold and the complex hyperbolicity of its compatible almost complex structures. I will present the interplay between these two notions. Then, I'll explain how this analysis allows one to get stability results for non complex hyperbolicity under deformation of the almost complex structure among the set of structures compatible to a fixed symplectic structure, thus getting a generalisation of Bangert's result which addressed the particular case of the standard torus.