We consider one-dimensional random walks in random environment which are
transient to the right. Our main interest is in the study of the sub-ballistic
regime, where at time $n$ the particle is typically at a distance of order
$O(n^\kappa)$ from the origin, $\kappa\in(0,1)$. We investigate the
probabilities of moderate deviations from this behaviour. Specifically, we are
interested in quenched and annealed probabilities of slowdown (at time $n$, the
particle is at a distance of order $O(n^{\nu_0})$ from the origin, $\nu_0\in
(0,\kappa)$), and speedup (at time $n$, the particle is at a distance of order
$n^{\nu_1}$ from the origin, $\nu_1\in (\kappa,1)$), for the current location
of the particle and for the hitting times. Also, we study probabilities of
backtracking: at time $n$, the particle is located around $(-n^\nu)$, thus
making an unusual excursion to the left. For the slowdown, our results are
valid in the ballistic case as well.