  Higher twisted $K$-theory is an extension of twisted $K$-theory introduced by
Ulrich Pennig which captures all of the homotopy-theoretic twists of
topological $K$-theory in a geometric way. We give an overview of his
formulation and key results, and reformulate the definition from a topological
perspective. We then investigate ways of producing explicit geometric
representatives of the higher twists of $K$-theory viewed as cohomology classes
in special cases using the clutching construction and when the class is
decomposable. Atiyah-Hirzebruch and Serre spectral sequences are developed and
information on their differentials is obtained, and these along with a
Mayer-Vietoris sequence in higher twisted $K$-theory are applied in order to
perform computations for a variety of spaces.
