On the Gromov hyperbolicity of domains in ℂ n

— We present several recent results dealing with the metric properties of domains in the complex Euclidean space Cn. We provide with examples of domains, endowed with Finsler or Kähler metrics, that are (or are not) Gromov hyperbolic. We also present how different notions, such as the Gromov hyperbolicity, the holomorphic bisectional curvature or the d’Angelo type may be related for smooth bounded domains in Cn. We also present several open questions in the core of the paper.