Dirac's ¿-matrices were introduced for the description of the behavior of electrons. As observed by D. Hestenes, the real algebra generated by these matrices is a faithful representation of the real Clifford algebra of Minkowski space-time M as a full matrix algebra. Consequently, it is possible to give a matrix-free formulation of Dirac's equations. In this paper we present a survey of algebraic structures of the real Clifford algebra of M. Current field and particle theories can be described concisely and conveniently by use of such structures. Here we present the algebraic aspects of the results of Hestenes in a structured and ordered way, and we add a number of extensions. Our treatment is basis-invariant. We emphasize that many of the algebraic structures depend on the choice of an orientation of M, on the choice of a timelike vector e, and on the choice of a spacelike vector ¿ orthogonal to e.