Integrated Semigroups and $C$-Semigroups and Their Applications

In this survey, some interesting generalizations of the theory of $C_0$-semigroups and their applications are presented. The survey is divided into three parts: "Integrated semigroups", "$C$-semigroups", and "Applications". Different approaches to the theory are discussed. The content presented is taken from articles of the last 20 years and also contains some results of the authors. Various applications are presented; the most of them concerns ill-posed Cauchy problems.