Abstract: 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.