  We solve an open problem by constructing quantum walks that not only detect
but also find marked vertices in a graph. In the case when the marked set $M$
consists of a single vertex, the number of steps of the quantum walk is
quadratically smaller than the classical hitting time $HT(P,M)$ of any
reversible random walk $P$ on the graph. In the case of multiple marked
elements, the number of steps is given in terms of a related quantity
$HT^+(\mathit{P,M})$ which we call extended hitting time.
  Our approach is new, simpler and more general than previous ones. We
introduce a notion of interpolation between the random walk $P$ and the
absorbing walk $P'$, whose marked states are absorbing. Then our quantum walk
is simply the quantum analogue of this interpolation. Contrary to previous
approaches, our results remain valid when the random walk $P$ is not
state-transitive. We also provide algorithms in the cases when only
approximations or bounds on parameters $p_M$ (the probability of picking a
marked vertex from the stationary distribution) and $HT^+(\mathit{P,M})$ are
known.
