This talk presents the development of the theory named in the title, during the last three years. Bifurcations in generic one-parameter families were classified; the answer appeared to be quite unexpected. An important and non-trivial question "who bifurcates?" was answered. A countable and even continuous number of topologically non-equivalent germs of bifurcation diagrams was found. Various structurally unstable three-parameter families were constructed. These are joint results of the speaker and his collaborators: N. Goncharuk, D. Filimonov, Yu. Kudryashov, N. Solodovnikov, I. Schurov and others. Some open problems will be stated.
All the necessary definitions will be given during the talk.