  Twisted complex $K$-theory can be defined for a space $X$ equipped with a
bundle of complex projective spaces, or, equivalently, with a bundle of
C$^*$-algebras. Up to equivalence, the twisting corresponds to an element of
$H^3(X;\Z)$. We give a systematic account of the definition and basic
properties of the twisted theory, emphasizing some points where it behaves
differently from ordinary $K$-theory. (We omit, however, its relations to
classical cohomology, which we shall treat in a sequel.) We develop an
equivariant version of the theory for the action of a compact Lie group,
proving that then the twistings are classified by the equivariant cohomology
group $H^3_G(X;\Z)$. We also consider some basic examples of twisted $K$-theory
classes, related to those appearing in the recent work of
Freed-Hopkins-Teleman.
