It has been known for some time that the value &Lgr; of the vacuum energy density affects the propagation equation for gravitons — the analogue of photons for the gravitational field. (For historical reasons, &Lgr; is also called “cosmological constant”.) More precisely, if &Lgr; is not zero, then a mass term appears in the propagation equation, such thatm2=−&Lgr;. As a consequence, the polarization states of gravitons also change, because a massless particle has only two polarization states while a massive particle has more. This effect of the &Lgr;‐term has been confirmed by recent calculations in a curved background, which is actually the only proper setting, since solutions of the classical Einstein equations in the presence of a &Lgr;‐term represent a space with constant curvature. A real value for the mass (when &Lgr;<0) will show up as a slight exponential damping in the gravitational potential, which is however strongly constrained by astronomical data. The consequences of an imaginary mass (for &Lgr;>0) are still unclear; on general grounds, one can expect the onset of instabilities in this case. This is also confirmed by numerical simulations of quantum gravity which became recently available. These properties gain a special interest in consideration of the following. (1) The most recent cosmological data indicate that &Lgr; is positive and of the order of 0.1 J/m3. Is this value compatible with a stable propagation of gravitons? (2) The answer to the previous question lies perhaps in the scale dependence of the effective value of &Lgr;. It could then happen that &Lgr; is actually negative at the small distance/large energy scale at which the quantum behavior of gravitational fields and waves becomes relevant. Applications for an advanced propulsion scheme is that local contributions to the vacuum energy density (remarkably in superconductors in certain states, and in very strong static electromagnetic fields) can change locally the sign of &Lgr;, and so affect locally the propagation and the properties of gravitons. The graviton wavefunction, for different values of the parameters, may be characterized by superluminal phase velocity or by unitarity only in imaginary valued time. This may indicate a connection between gravitons and Faster‐Than‐Light travel. © 2004 American Institute of Physics