  We study virtual levels of $N$-particle Schr\"odinger operators and prove
that if the particles are one-dimensional and $N\ge 3$, then virtual levels at
the bottom of the essential spectrum correspond to eigenvalues. The same is
true for two-dimensional particles if $N\ge 4$. These results are applied to
prove the non-existence of the Efimov effect in systems of $N\ge 4$
one-dimensional or $N\ge 5$ two-dimensional particles.
