Although the study of accelerated motion is necessary in Special Relativity, as a rule, in standard textbooks little attention is payed to this subject. Perhaps this is due to the difficulty of the comprehension of “acceleration” in spacetime and its involved “behavior” under the Lorentz transformation. Indeed, with the exception of the proper frame, the zeroth component of the acceleration four-vector enters in the spatial part creating confusion. Furthermore the Lorentz transformation of the acceleration four-vector does not reveal a clear kinematic role for the temporal and the spatial parts. However the extensive study of four-acceleration is necessary, because it completes our understanding of relativistic kinematics, relates kinematics with the dynamics and finally it takes Special Relativity over to General Relativity in a natural way. In addition the four-acceleration finds application in many physical phenomena, such as, the radiation of an accelerated charge, the annihilation of anti-proton, the resonances of strange particles etc.