  We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d.\
random field of obstacles. When the particle hits an obstacle, it disappears
with a positive probability. We obtain quenched and annealed bounds on the
tails of the survival time in the general $d$-dimensional case. We then
consider a simplified one-dimensional model (where transition probabilities and
obstacles are independent and the RWRE only moves to neighbour sites), and
obtain finer results for the tail of the survival time. In addition, we study
also the "mixed" probability measures (quenched with respect to the obstacles
and annealed with respect to the transition probabilities and vice-versa) and
give results for tails of the survival time with respect to these probability
measures. Further, we apply the same methods to obtain bounds for the tails of
hitting times of Branching Random Walks in Random Environment (BRWRE).
