Conformal, quasiconformal, topological, postcritical and Julia structural stability for dynamics of iteration of holomorphic functions and their relationships, as presented by Sullivan and McMullen in 1998, based on the work of Mañé, Sullivan and Sad in 1983. Genericity of structural stability. Relationship of structural stability with hyperbolicity. Notion of invariant line field on a Julia set, of Sullivan and McMullen in 1993, and its application to obtain a necessary and sufficient condition for a quasiconformal conjugacy to be holomorphic. Structural unstability of rational functions with a Herman ring, as by Mañé in 1985