In the past 30 years the theory of quantum Markov chains (QMC) has undergone several developments. The attempt to give an intrinsic operator–theoretical characterization of QMC produced a deep analysis, due to C. Cecchini, of various notions of quantum Markovianity. The notion of Markovianity on CAR algebras, and the corresponding structure theorems, revealed a surprisingly richer structure than in the infinite tensor product case. Moreover, as it often happens in quantum probability, the efforts to better understand the quantum case has lead to question some deeply rooted beliefs concerning classical Markov processes. Applications to physics have proliferated in several different directions, ranging from the interesting theoretical results of Fannes, Nachtergaele, Werner, Matsui, Mohari, Mukhammedov, Ohno,... to numerical simulations related to the Bethe approximation. The results of Lindblad, Alicki and Fannes on the notion of quantum dynamical entropy and the subsequent extension by Ohya, Watanabe and others, have established a connection between QMC and the theory of quantum chaos. Finally Petz and his school has shown that QMC play a relevant role also in quantum information, notably in the problems related to the capacity of quantum channels. Now the boundary of this line of research is the extension of the above results to Markov fields (i.e. processes with multidimensional index set). What is needed is not so much an abstract theory (many variants are possible and some of them already published) as a new nontrivial, class of concrete examples which could play for fields a role analogue to that, played in the 1–dimensional case, by the QMC, i.e. a benchmark on which to test the power of different theoretical proposals. Now such class of examples has begun to be developed. I will use this class to illustrate some points of the abstract theory and some interesting open problems.