Lyapunov’s theorem states that the range of a nonatomic vector-valued measure (with values in \(\mathbb{C}^ n\)) is a compact convex set. The volume under review contains a comprehensive study of generalizations of Lyapunov’s result in the following form. Let \(\Psi\) be an affine map of a convex set \(Q\) in a linear space \(X\) into another linear space \(Y\). By a theorem of Lyapunov’s type the authors mean that for each \(x\in Q\) there is an extreme point \(e\in Q\) such that \(\Psi(e)\) is close to \(\Psi(x)\). In the typical case \(X\) is a von Neumann algebra, \(Q\) is a weak\(^*\) closed face and \(Y\) is \(\mathbb{C}^ n\) or \(M_ n(\mathbb{C})\). When the range space is noncommutative (i.e., a matrix algebra) the analysis is much harder and the authors make several conjectures.