  Non reciprocal spin waves have a chiral asymmetry so that their energy is
different for two opposite wave vectors. They are found in atomically thin
ferromagnetic overlayers with in plane magnetization and are linked to the
anti-symmetric Dzyaloshinskii-Moriya surface exchange. We use an itinerant
fermion theory based on first principles calculations to predict that
non-reciprocal magnons can occur in Fe$_3$GeTe$_2$, the first stand alone
metallic two dimensional crystal with off-plane magnetization. We find that
both the energy and lifetime of magnons are non-reciprocal and we predict that
acoustic magnons can have lifetimes up to hundreds of picoseconds, orders of
magnitude larger than in other conducting magnets.
