In Section 1, the authors establish the models of two kinds of Markov chains
in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTRE(+) by an initial distribution Φ and a random Markov kernel (RMK) p(γ). In Section 3, the authors establish several equivalence theorems on MCSTRE and MCSTRE(+). Finally, the authors give two very important examples of MCMSTRE, the random walk in spce-time random environment and the Markov branching chain in space-time random environment.