We prove the Gromov non-hyperbolicity with respect to the Kobayashi distance for C1,1-smooth convex domains in C2 which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. The same is shown for “generic” product spaces, as well as for the symmetrized polydisc and the tetrablock. On the other hand, examples of smooth, non-pseudoconvex, Gromov hyperbolic domains in Cn are given.