Representing the relativistic physical fields as sections of the Clifford Bundle (or of the Spin-Clifford Bundle) of Minkowski spacetime we show that all the relativistic wave equations satisfied by these fields possess solutions traveling with arbitrary speeds o ≤ υ < ∞. By giving rigorous mathematical definitions of reference frames and of the Principle of Relativity (PR) we prove that physical realizations of the υ > 1 solutions of, e.g., the Maxwell equations imply in a breakdown of the PR, but in no contradiction at all with known physical facts. © VSP 1996.44355400Rodrigues Jr., W.A., De Souza, Q.A.G., The clifford bundle and the nature of the gravitational field (1993) Found, of Phys., 23, pp. 1465-1490Rodrigues Jr., W.A., De Souza, Q.A.G., Vaz Jr., J., Spinor fields and superfields as equivalence classes of exterior algebra fields (1995) Clifford Algebras and Spinor Structures, pp. 177-198. , R. Ablamowicz and P. Lounesto (eds.), Kluwer Acad. 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