The chapter is devoted to the single topic of extending $\mathscr C$-valued operators. Its first section presents the global approach to the extension of operators: Zippin’s characterisation of $\mathscr C$-trivial embeddings by means of weak*-continuous selectors and a few noteworthy applications. The second section presents the Lindenstrauss-Pe\l czy\’nski theorem with two different proofs: the first one combines homological techniques with the global approach, while the second is Lindenstrauss-Pe\l czy\’nski’s original proof. The analysis of their proof is indispensable for understanding Kalton’s imaginative inventions that lead to the so-called $L^*$ and $m_1$-type properties and to a decent list of $\mathscr C$-extensible spaces. The next two sections contain, respectively, those points of the Lipschitz theory that are necessary to develop the linear theory and different aspects of Zippin’s problem: which separable Banach spaces $X$ satisfy $\operatorname{Ext}(X, C(K))=0\,$? The problem admits an interesting gradation in terms of the topological complexity of $K$. The final section reports the complete solution of the problem of whether $\operatorname{Ext}(C(K), c_0)\neq 0$ for all non-metrisable compacta $K$.

This chapter focuses on the possibility of extending isomorphisms or isometries to maps of the same type. It presents all known results about the automorphic space problem of Lindenstrauss and Rosenthal, including a dichotomy theorem, and about spaces of universal disposition already envisioned by Gurariy. It also treats finite-dimensional variations of those properties: the rich theory of UFO spaces and finitely automorphic quasi-Banach spaces. The topics of how many positions a Banach space can occupy in a bigger superspace and how many twisted sums of two spaces exist are considered.

The final chapter of the book returns to the place the journey started: classical Banach space theory, with a twist. We can now provide solutions, or at least a better understanding, for a number of open problems. Among the topics covered, the reader will encounter vector-valued forms of Sobczyk’s theorem, isomorphically polyhedral $\mathscr L_\infty$-spaces, Lipschitz and uniformly homeomorphic $\mathscr L_\infty$-spaces, properties of kernels of quotient operators from $\mathscr L_1$-spaces, sophisticated 3-space problems, the extension of $\mathscr L_\infty$-valued operators, Kadec spaces, Kalton-Peck spaces and, at last, the space $Z_2$. All these topics can be easily considered as part of classical Banach space theory, even if the techniques we employ involve most of the machinery developed throughout the book.