Abstract: Unparticles as recently suggested by Georgi are identities that are not constrained by dispersion relations (and are thus not particles) but are governed by their scaling dimension. We show that their coupling to particles results in novel long-range forces between particles that are totally different from known ones or those arising in any other new physics contemplated so far. If such forces were discovered in future precision measurements it would point to unparticle physics unambiguously, and thus impact significantly the basic concepts of particle physics. On the other hand, we use the existing precisely measured long-range spin-spin interaction of electrons to set stringent bounds on the unparticle coupling to the electron. We also mention briefly the prospects for research in this field.