Several experiments performed in recent years are claimed to have demonstrated superluminal wave motion. A major aim of this book is to elucidate what is really meant and implied by superluminal wave motion - a question that has caused much confusion until now. Using modern mathematical tools, the authors demonstrate that none of the results imply any violation of the principle of relativity. They also clarify the meaning of several types of superluminal velocity that appear in quantum theory; for example, the Hegerfeldt paradox and formalism is given in sufficient detail that students will also be able to follow the arguments. It is proved that all linear relativistic wave equations of mathematical physics (Maxwell, Dirac, Weyl, the homogeneous wave equation and the Klein-Gordon equation) possess arbitrary velocities solutions, which are undistorted progressive waves.