This paper is intended to present the basic properties of KOtheory for real C * -algebras and to explain its relationship with complex Ktheory and with KR-theory. Whenever possible we will rely upon proofs in printed literature, particularly the work of Karoubi [36], Wood [64], Schröder [54], and more recent work of Boersema [14] and J. M. Rosenberg [49]. In addition, we shall explain how KO-theory is related to the Ten-Fold Way in physics and point out how some deeper features of KO-theory for operator algebras may provide powerful new tools there. Commutative real C * -algebras not of the form C R (X) will play a special role. We also will identify Atiyah's KR 0 (X, τ )) in terms of KO 0 of an associated real C * -algebra.Unfortunately, there is no single reference for KO-theory for operator algebras that begins to compare with Blackadar's wonderful exposition of complex K-theory [13]. This paper is intended to provide a platform upon which mathematicians and mathematical physicists can rely in order to use these new tools in their research.14 Suppose that xo is the base-point of X andThis implies that α * : X → Y preserves base-points. When we say that f * ≃ g * : C R (Y ) → C R (X) are base-point preserving homotopic, we mean that at each stage the homotopy must preserve base-points.