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The R-matrix formalism for the construction of integrable systems withinfinitely many degrees of freedom is reviewed. Its application to Poisson,noncommutative and loop algebras as well as central extension procedure arepresented. The theory is developed for (1+1)-dimensional case where the spacevariable belongs either to R or to various discrete sets. Then, the extensiononto (2+1)-dimensional case is made, when the second space variable belongs toR. The formalism presented contains many proofs and important details to makeit self-contained and complete. The general theory is applied to severalinfinite dimensional Lie algebras in order to construct both dispersionless anddispersive (soliton) integrable field systems.

Classical R-matrix theory for bi-Hamiltonian field systems