Before we start with the main topic of this book, we will recall some important aspects of the theory of \(C_0\) -semigroups. There are several monographs on \(C_0\) -semigroups and for more details we refer the reader from example to the books by Engel and Nagel [101], Goldstein [119], Pazy [197], Arendt, Batty, Hieber and Neubrander [17] or Bátkai, Kramar Fijavž and Rhandi [33], just to mention a few. This chapter is supposed to provide an overview of the theory of \(C_0\) -semigroups, as this is also crucial to understand the theory of bi-continuous semigroups. However, in order to emphasize the introductory character of this chapter, we will only state the important results without proofs. For more details we then refer to the references mentioned beforehand.