Abstract: Almost the whole of the rich theory of Markov processes is concerned with the special case in which the transition probabilities are time invariant, so that the Kolmogorov differential equations are autonomous. The conventional wisdom is that there are few interesting results without this restriction, but in applications the invariance assumption is often unrealistic. It is therefore worth asking whether some of the deeper results of classical Markov theory can be formulated so as to remain true in the general case. This will be illustrated in relation to the theorem of Ornstein that each transition probability in a process with a countable infinity of states is either always or never zero.