Markov property: recent developments on the Quantum Markov property

We review recent developments in the theory of quantum Markov states on the standard $\bz^{d}$--spin lattice. A Dobrushin theory for quantum Markov fields is proposed. In the one--di\-men\-sional case where the order plays a crucial role, the structure arising from a quantum Markov state is fully understood. In this situation we obtain a splitting of a Markov state into a classical part, and a purely quantum part. This result allows us to provide a reconstruction theorem for quantum Markov states on chains