  This paper is intended to present the basic properties of $KO$-theory for
real $C^*$-algebras and to explain its relationship with complex $K$-theory and
with $KR$- theory. Whenever possible we will rely upon proofs in printed
literature, particularly the work of Karoubi, Wood, Schr\"oder, and more recent
work of Boersema and J. M. Rosenberg. 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 $KR^0(X, \tau ))$ in terms of
$KO_0$ of an associated real $C^*$-algebra.
