  We discuss twisted cohomology, not just for ordinary cohomology but also for
$K$-theory and other exceptional cohomology theories, and discuss several of
the applications of these in mathematical physics. Our list of applications is
by no means exhaustive, but we are hoping that it is extensive enough to give
the reader a feel for the possible applications of twisted theories in many
different contexts. We also give many suggestions for further reading, but this
subject has now expanded to the point where the bibliography is necessarily
very incomplete.
