The possibility of mass in the context of scale-invariant, generally covariant theories, is discussed. Scale invariance is considered in the context of a gravitational theory where the action, in the first order formalism, is of the formS=∫L1&Fgr;d4x+∫L2−gd4xwhere &Fgr; is a density built out of degrees of freedom independent of the metric. For global scale invariance, a “dilaton” &fgr; has to be introduced, with non-trivial potentialsV(&fgr;)=f1e&agr;&fgr;inL1andU(&fgr;)=f2e2&agr;&fgr;inL2.This leads to non-trivial mass generation and a potential for &fgr; which is interesting for new inflation. Scale invariant mass terms for fermions lead to a possible explanation of the present day accelerated universe and of cosmic coincidences. Although the scale symmetry is spontaneously broken there is no Goldstone boson. This surprising effect is due to the fact that in spite of the fact that there is a locally conserved current, no globally conserved dilatation charge exists due to the singular infrared behavior of the spatial components of such a current. ©1999 American Institute of Physics.