The Dirac equation was formulated by Paul Dirac in 1929, and ever since then it has played an important role in several areas of physics and mathematics and has plenty of applications in quantum mechanics. The equation describes the relativistic spin-1/2 particles, such as, electrons or positrons, which move in different electric and magnetic fields. In this book, some topics concerning the Dirac equation are studied. Since the self-adjointness of operators is a fundamental information in quantum mechanics, several self-adjoint extensions of the Dirac operator coupled to electromagnetic potentials are constructed by means of Hardy-type estimates. On the other hand, the Dirac equation can be seen as a dispersive model, therefore, some Strichartz estimates are studied, and in particular, counterexamples for Strichartz estimates for the solution of the Dirac equation are constructed. This book, which is the publication of the author's PhD dissertation, is suitable for graduate students and researchers in mathematics interested in the Dirac equation.