In this chapter, we focus on the study of intermediate and extrapolation spaces associated with bi-continuous semigroups. Building upon the foundational results introduced in earlier chapters, we explore how these spaces contribute to a deeper understanding of the generators of bi-continuous semigroups. In particular, we revisit the Hille–Yosida generation theorem for bi-continuous semigroups and present an alternative proof using the structure of extrapolation spaces. This chapter not only extends the classical semigroup theory but also opens up new perspectives for applications in settings where bi-continuity is essential.