  We introduce the concept of a deterministic walk in a deterministic
environment on a countable state space (DWDE). For the deterministic walk in a
fixed environment we establish properties analogous to those found in Markov
chain theory, but for systems that do not in general have the Markov property.
In particular, we establish hypotheses ensuring that a DWDE on $\Z$ is either
recurrent or transient. An immediate consequence of this result is that a
symmetric DWDE on $\Z$ is recurrent. Moreover, in the transient case, we show
that the probability that the DWDE diverges to $+ \infty$ is either 0 or 1. In
certain cases we compute the direction of divergence in the transient case.
