  We introduce the concept of twisted contact groupoids, as an extension either
of contact groupoids or of twisted symplectic ones, and we discuss the
integration of twisted Jacobi manifolds by twisted contact groupoids. We also
investigate the very close relationships which link homogeneous twisted Poisson
manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic
groupoids with twisted contact ones. Some examples for each structure are
presented.
