  We present analytical results for the distribution of first hitting times of
non-backtracking random walks on finite Erd\H{o}s-R\'enyi networks of $N$
nodes. The walkers hop randomly between adjacent nodes on the network, without
stepping back to the previous node, until they hit a node which they have
already visited before or get trapped in a dead-end node. At this point, the
path is terminated. The length, $d$, of the resulting path, is called the first
hitting time. Using recursion equations, we obtain analytical results for the
tail distribution of first hitting times, $P(d>\ell)$, $\ell=0,1,2,\dots$, of
non-backtracking random walks starting from a random initial node. It turns out
that the distribution $P(d>\ell)$ is given by a product of a discrete Rayleigh
distribution and an exponential distribution. It is found that the paths of
non-backtracking random walks, up to their termination at the first hitting
time, are longer, on average, than those of the corresponding simple RWs.
However, they are shorter than those of SAWs on the same network, which
terminate at the last hitting time. We obtain analytical results for the
probabilities, $p_{\rm ret}$ and $p_{\rm trap}$, that a path will terminate by
retracing, namely stepping into an already visited node, or by trapping, namely
entering a node of degree $k=1$, which has no exit link, respectively. It is
shown that in dilute networks the dominant termination scenario is trapping
while in dense networks most paths terminate by retracing. We obtain
expressions for the conditional tail distributions of path lengths,
$P(d>\ell|{\rm ret})$ and $P(d>\ell|\rm{trap})$, for those paths which
terminate by retracing or by trapping, respectively. We also study a class of
generalized non-backtracking random walk models which not only avoid the
backtracking step into the previous node but avoid stepping into the last $S$
visited nodes.
