On the approaching geodesics property

We survey some recent results and open questions on the approaching geodesics property and its application to the study of the Gromov and horofunction compactifications of a proper geodesic Gromov metric space. We obtain results on the dynamics of isometries and we exhibit an example of a Gromov hyperbolic domain of $\mathbb{C}$ which does not satisfy the approaching geodesic property.