A review of some recent results on the dynamical theory of the Yang-
Baxter maps (also known as set-theoretical solutions to the quantum Yang-Baxter
equation) is given. The central question is the integrability of the transfer dynamics.
The relations with matrix factorisations, matrix KdV solitons, Poisson Lie groups, geometric
crystals and tropical combinatorics are discussed and demonstrated on several
concrete examples.