In this overview talk will deal with generalizations of Herglotz-Nevanlinna functions . We will start with some updates concerning functions in several variables and then consider generalizations (still in only one variable though) that are not any more related to passive systems. Particular focus will be put on quasi-Herglotz functions, these are complex linear conbinations of usual Herglotz functions and they exhibit interesting phenomena. (Part of the talk is based on joint work with Mitja Nedic).