  The hierarchy of commuting maps related to a set-theoretical solution of the
quantum Yang-Baxter equation (Yang-Baxter map) is introduced. They can be
considered as dynamical analogues of the monodromy and/or transfer-matrices.
The general scheme of producing Yang-Baxter maps based on matrix factorisation
is discussed in the context of the integrability problem for the corresponding
dynamical systems. Some examples of birational Yang-Baxter maps coming from the
theory of the periodic dressing chain and matrix KdV equation are discussed.
