Hypothetical particles, called tachyons, which travel faster than the speed of light, are studied both as classical particles subject to the mathematical restrictions of special relativity theory and as particles described by a many-particle quantum field quantized according to Fermi-Dirac statistics. Negative-energy particles appear in the theory but are interpreted physically as positive-energy particles with the opposite momentum. We find that the total number of particles present is not Lorentz invariant, but (for charged tachyons) the total charge is. It is possible to contrive causal, paradoxes leading to causal loops (effect precedes cause) but such difficulties can be remedied by defining a cosmological preferred frame and postulating that emission and absorption of tachyons can only take place between members of some restricted class of observers with velocities less than some critical value relative to the preferred inertial reference frame. The causality of tachyon signal transmission is considered in terms of entropy flux, and we find that we must reinterpret signals carrying positive entropy backward in time as signals carrying negative entropy forward in time. Several experiments have been conducted to find tachyons, both charged and neutral, but all have been unsuccessful.