  We study the biased random walk where at each step of a random walk a
"controller" can, with a certain small probability, move the walk to an
arbitrary neighbour. This model was introduced by Azar et al. [STOC'1992]; we
extend their work to the time dependent setting and consider cover times of
this walk. We obtain new bounds on the cover and hitting times. Azar et al.
conjectured that the controller can increase the stationary probability of a
vertex from $p$ to $p^{1-\epsilon}$; while this conjecture is not true in full
generality, we propose a best-possible amended version of this conjecture and
confirm it for a broad class of graphs.
  We also consider the problem of computing an optimal strategy for the
controller to minimise the cover time and show that for directed graphs
determining the cover time is PSPACE-complete.
