High-energy density beam processes for welding, including laser beam welding and electron beam welding, are essential processes in many industries and provide unique characteristics that are not available with other processes used for welding. More recently, these high-energy density beams have been used to great advantage for additive manufacturing. This review of the literature serves to provide an overview of the evolution of the laser and electron beam processes including the fundamental nature of the beam itself and how such a high-energy density beam has been applied for welding. The unique nature of each process regarding weld bead formation and penetration, and metallurgical considerations are covered in detail. In addition, the evolution of monitoring systems for both characterization and control of these beam processes is reviewed. Over 500 references have been cited in this comprehensive review that will allow the reader to both understand the current state-of-the-art and explore in more detail the fundamental concepts and practical applications of these processes.