According to the uniformisation theorem, in complex dimension one, there is an intimate connection between complex and hyperbolic structures. However, in higher dimensions, the two geometries diverge. We will discuss both the existence of almost complex structures on hyperbolic manifolds, and the potential obstructions to their integrability. In odd complex dimensions the discussion generalizes to certain stably parallelizable manifolds, of which the hyperbolic ones constitute a very special subclass. This seminar is supported by PRIN 20228JRCYB - CUP B53D23009140006.