  We study the stationary properties as well as the non-stationary dynamics of
the one-dimensional partially asymmetric exclusion process with position
dependent random hop rates. In a finite system of $L$ sites the stationary
current, $J$, is determined by the largest barrier and the corresponding
waiting time, $\tau \sim J^{-1}$, is related to the waiting time of a single
random walker, $\tau_{rw}$, as $\tau \sim \tau_{rw}^{1/2}$. The current is
found to vanish as: $J \sim L^{-z/2}$, where $z$ is the dynamical exponent of
the biased single particle Sinai walk. Typical stationary states are phase
separated: At the largest barrier almost all particles queue at one side and
almost all holes are at the other side. The high-density (low-density) region,
is divided into $\sim L^{1/2}$ connected parts of particles (holes) which are
separated by islands of holes (particles) located at the subleading barriers
(valleys). We also study non-stationary processes of the system, like
coarsening and invasion. Finally we discuss some related models, where
particles of larger size or multiple occupation of lattice sites is considered.
