  We investigate the statistics of the first detected passage time of a quantum
walk. The postulates of quantum theory, in particular the collapse of the wave
function upon measurement, reveal an intimate connection between the wave
function of a process free of measurements, i.e. the solution of the
Schr\"odinger equation, and the statistics of first detection events on a site.
For stroboscopic measurements a quantum renewal equation yields basic
properties of quantum walks. For example, for a tight binding model on a ring
we discover critical sampling times, diverging quantities such as the mean time
for first detection, and an optimal detection rate. For a quantum walk on an
infinite line the probability of first detection decays like
$(\mbox{time})^{-3}$ with a superimposed oscillation, critical behavior for a
specific choice of sampling time, and vanishing amplitude when the sampling
time approaches zero due to the quantum Zeno effect.
