We study the motion of two non-interacting quantum
particles performing a random walk on a line and analyze the
probability that the two particles meet after a certain number of
steps (meeting problem). The results are compared with the
corresponding classical problem and differences are pointed out.
Analytic formulas for the meeting probability and asymptotic
behavior are derived. Moreover, quantum random walk with more
particles brings further possibilities unavailable for the
classical random walk - the walkers can be indistinguishable
bosons or fermions and they can be initially entangled. The effect
of this additional freedom on the meeting probability is analyzed.