In this chapter, which is original and initiates the research part of the book, we introduce a particular strongly continuous semigroup and will show that it can be considered in some sense a “universal” semigroup, since it allows to reproduce any mild solution. This is an important result not only on its own but also, for example, because it may well happen that we can prove a certain property of the mild solution for the given shift semigroup, but not in general for arbitrary semigroups.