This chapter introduces key approximation techniques for bi-continuous semigroups, focusing on the foundational results that allow for effective handling of these semigroups as we will point out in some examples, see Sect. 6.3. The Trotter–Kato approximation theorems form the basis, offering precise conditions under which convergence of generators (in a certain sense) of the convergence of resolvents imply convergence of the associated bi-continuous semigroups, see Theorems 6.3 and 6.6. This framework is further enriched by the Chernoff product formula (Theorem 6.8), which enables the approximation of semigroups via compositions of simpler operators. We also examine the Post–Widder inversion formula, cf. Corollary 6.9, providing a method to reconstruct semigroups from Laplace transforms as well as the Lie–Trotter product formula which gives a first taste for perturbations, cf. Corollary 6.10. Together, these methods form a versatile toolkit for approximating and analyzing bi-continuous semigroups, with broad implications in the study of evolution equations.