We examine arguments in favour of tachyons and extensions of the Lorentz Group to include superluminal boosts. We offer support for the existence of tachyons from a recent finding in classical electromagnetic theory. With a slight modification to the standard theory we observe that a single superluminal charge will bind to itself in a self-sustaining circular orbit, suggestive of a possible (modified) electromagnetic interpretation of hadrons. Symmetries in that theory are used in the subsequent analysis as a starting point in the search for extensions of the Lorentz Group. We discuss the issue of real versus imaginary coordinates in superluminal transformations of frame, and point out that the Lie group extended to include superluminal boosts is non-connected. We close with some speculation on the implications for faster-than light travel.