A survey on linear (additive) preserver problems

This survey is articulated around two major axis. The first one concerns the Kaplansky problem; the history of the problem and several results are presented. The second one concerns some new preserver problems (concerning the generalized inverse, Fredholm or semi-Fredholm operators). The common point of these results is that they are interesting only in the infinite dimensional situation. Several open questions are mentioned over all the paper.