The eight-component (8-C) relativistic wave equation for spin-1/2 particles was developed by B. A. Robson and D. S. Staudte in 1993 using the procedure analogous to the one used earlier on the Klein-Gordon equation. The relativistic covariance and the solution behaviour of the 8-C equation have been studied extensively by D. S. Staudte in 1993. The book provides the study of the 8-C equation, particularly its application to physical problems such as Compton scattering and transition probabilities in hydrogenic atoms. The application of the 8-C equation to the Compton scattering problem shows that it gives the same well-known Klein-Nishina cross section formula as that initially obtained using the Dirac equation. The 8-C equation has also been applied to the calculation of the transition probabilities for the components of both the Balmer and Lyman alpha-lines of hydrogenic atoms, and the results show the first indication that the 8-C equation and the Dirac equation are actually not equivalent. This book is intended for those who are interested in the comparative study between the 8-C equation and the well-known Dirac equation, as well as their theoretical consequences.