<p>We deal with the analogous of the so-called Quantum Markov Chains for Fermi systems (i.e. the models whose fundamental fields enjoy the Canonical Anti-commutation Relations), and discuss their main properties. Since the involution and the product highly differ from those of the usual tensor product, the positivity of the involved operators and functionals, not automatically satisfied, is a very delicate question. As a consequence, the positivity should be checked directly for the specific models under consideration. Therefore, we analyse some pivotal cases, providing examples which go beyond the simpler ones already studied in literature. As an interesting result, we prove that the symmetric states, that is those which are invariant under the natural action of the group of all finite permutations, are described in terms of particular Quantum Markov Chains.</p>