  Quantum backflow is the classically-forbidden effect pertaining to the fact
that a particle with a positive momentum may exhibit a negative probability
current at some space-time point. We investigate how this peculiar phenomenon
extends to many-particle systems. We give a general formulation of quantum
backflow for systems formed of $N$ free nonrelativistic structureless
particles, either identical or distinguishable. Restricting our attention to
bosonic systems where the $N$ identical bosons are in the same one-particle
state allows us in particular to analytically show that the maximum achievable
amount of quantum backflow in this case becomes arbitrarily small for large
values of $N$.
