In this chapter we plunge into the non-linear aspects of the theory of twisted sums. One of the objectives of this chapter is to provide the reader with practical ways to construct non-trivial exact sequences $0 \longrightarrow Y \longrightarrow \cdot \longrightarrow X \longrightarrow 0$ when only the spaces $Y$ and $X$ are known. The central idea here is that such exact sequences correspond to a certain type of non-linear map called a quasilinear map $\Phi: X \longrightarrow Y$. The chapter has been organised so that the reader can reach at an early stage a number of important applications. The topics covered include finding pairs of quasi-Banach spaces $X, Y$ such that all exact sequences $0 \longrightarrow Y \longrightarrow \cdot \longrightarrow X \longrightarrow 0$ split, natural representations for the functor $\operatorname{Ext}$, getting valuable insight into the structure of exact sequences and twisted sum spaces, a duality theory for exact sequences of Banach spaces (including a non-linear Hahn-Banach theorem), uniform boundedness principles for exact sequences leading to a local theory for exact sequences, homological properties of the spaces $\ell_p$ and $L_p$, type of twisted sums, $\mathscr K$-spaces and the Kalton-Peck maps.

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.