"In the late 1970's, two fundamental concepts appeared in the theory of integrable systems, recursion operators (sometimes called strong symmetries or Λ-operators) that map symmetries to symmetries or, dually, conservation laws to conservation laws, and bi-Hamiltonian structures. The latter notion is due to Magri (1978) though Gelfand and Dorfman introduced it independently a year later under the name of `Hamiltonian pairs'. Examples of bi-Hamiltonian structures had been discovered by Adler, and by Gelfand and Dikii in a 1978 paper which was never published in Russian but was finally published, in English translation, in 1987 in the Collected Papers of Gelfand (volume 1) though it is still often cited as a preprint, as is the case in the book under review. The operator, obtained by composing one Hamiltonian mapping with the inverse of another (assumed to be invertible) is a recursion operator for any bi-Hamiltonian system, i.e. vector field which is Hamiltonian with respect to both structures. The Hamiltonian or Poisson structures of a Hamiltonian pair satisfy a compatibility condition which implies that the Nijenhuis torsion of this operator vanishes. This important fact and its applications are to be found in the first (1979) of the series of papers published by Gelfand and Dorfman in Funct. Anal. Appl. while it was discovered independently and at the same time by Magri and emphasized by Fokas and Fuchssteiner. From then on the theory of bi-Hamiltonian systems and its applications to integrable systems developed rapidly."