The theory of relativity forbids the superluminal travel of ordinary matter. However, it is possible to amend the theory of relativity and to develop a theory permitting superluminal travel. The acceptability of the features needed for superluminal travel is discussed.

Tolman's paradox arises in Lorentz-invariant theories of superluminal particles. In this paper we first try to clarify the nature of the paradox and what it means to solve it. We then analyze the various attempts made to either solve or eliminate it. We show that general consequences can be drawn which hold in essentially all paradox-free schemes proposed so far.

A tachyon or a superluminal wave group appears as a spacelike structure through a region in the reference frame in which it is at rest. Such a structure can arise from residue left (a) by a particle or wave group traveling at fundamental speed c or less, or (b) in the creation or separation of particles. Thus, Maund's argument does not prove that tachyons cannot exist. Creation may include lepton-quark production as well as particle-antiparticle production.