  For an integral cohomology class H of degree n+2 on a space X, we define
twisted Morava K-theory K(n)(X; H) at the prime 2, as well as an integral
analogue. We explore properties of this twisted cohomology theory, study a
twisted Atiyah-Hirzebruch spectral sequence, and give a universal coefficient
theorem (in the spirit of Khorami). We extend the construction to define
twisted Morava E-theory, and provide applications to string theory and
M-theory.
