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We introduce the concept of twisted contact groupoids, as an extension eitherof contact groupoids or of twisted symplectic ones, and we discuss theintegration of twisted Jacobi manifolds by twisted contact groupoids. We alsoinvestigate the very close relationships which link homogeneous twisted Poissonmanifolds with twisted Jacobi manifolds and homogeneous twisted symplecticgroupoids with twisted contact ones. Some examples for each structure arepresented.

On twisted contact groupoids and on integration of twisted Jacobi manifolds