Discrete local holomorphic dynamics

This is a survey on the local structure about a fixed point of discrete finite-dimensional holomorphic dynamical systems, discussing in particular the existence of local topological conjugacies to normal forms, and the structure of local stable sets in the non-hyperbolic case. The author hopes to keep the survey up to date, and thus it would be grateful to anybody pointing out missing (or mistaken) results and references, and/or suggesting topics to be included or expanded in it.