The present paper contributes significantly to the study of Lipschitz isomorphisms of Banach spaces. For a Banach space \(X\), let Lip\(_0(X)\) denote the space of Lipschitz functions on \(X\) that vanish at~\(0\), endowed with the Lipschitz constant as its canonical norm. This is a dual space whose canonical predual \({\mathcal F}(X)\) is spanned by the evaluation mappings \(\delta(x): f \mapsto f(x)\) in Lip\(_0(X)^*\). It is this space \({\mathcal F}(X)\) that the authors call the Lipschitz free space over \(X\) because of its functorial properties that are reminiscent of free groups; for example, every Lipschitz map from \(X\) to \(Y\) ``lifts'' to a linear operator from \({\mathcal F}(X) \) to \({\mathcal F}(Y)\). (The space \({\mathcal F}(X)\) has also been referred to as the Arens-Eells space in the literature.) In the second section, the key notions are introduced. For the nonlinear isometric mapping \(\delta: x \mapsto \delta(x)\), the authors construct a linear left inverse \(\beta\), i.e., a linear quotient map \(\beta: {\mathcal F}(X) \to X\) such that \(\beta \delta = \text{ Id}_X\). The crucial property is now whether \(X\) has the Lipschitz lifting property, LLP for short, meaning that there is a linear operator \(T: X\to {\mathcal F}(X)\) such that \(\beta T = \text{ Id}_X\); if \(T\) can be chosen isometric, this is called the isometric LLP. By construction, \({\mathcal F}(X)\) is a twisted sum of \(Z_X:= \ker \beta\) and \(X\), and the main theorem in Section~2 says the sum splits, i.e., \({\mathcal F}(X)\) is linearly isomorphic to \({\mathcal G}(X):= Z_X \oplus X\), if and only if \(X\) has the LLP; \({\mathcal F}(X)\) is, however, always Lipschitz isomorphic to \({\mathcal G}(X)\). One of the major problems in the Lipschitz theory of Banach spaces is to know whether separable Lipschitz isomorphic Banach spaces are actually linearly isomorphic. The previous theorem suggests candidates for counterexamples, but it is shown in Theorem~3.1 that every separable Banach space has the isometric LLP so that this approach doesn't yield counterexamples in the separable case. As a corollary, if \(F: X\to Y\) is a nonlinear isometry on a separable space \(X\), then \(Y\) contains a subspace linearly isometric to~\(X\). However, in the nonseparable case, which is radically different, there are lots of counterexamples; for example (Theorem~4.3), a WCG space has the LLP if and only if it is separable. Also, \(\ell_\infty\) fails the LLP. In the final section, it is shown that the bounded approximation property (BAP) is a Lipschitz invariant, since it is shown that the \(\lambda\)-BAP for \(X\) is equivalent to the \(\lambda\)-BAP for \({\mathcal F}(X)\) and also to the \(\lambda\)-Lipschitz-BAP for \(X\). Reviewer's remark: For further results along these lines, see \textit{N.~Kalton's} paper in [Collect.\ Math.\ 55, No. 2, 171--217 (2004)].