  In this paper we introduce the twistor space of a Riemannian manifold with an
even Clifford structure. This notion generalizes the twistor space of
quaternion-Hermitian manifolds and weak-Spin(9) structures. We also construct
almost complex structures on the twistor space for parallel even Clifford
structures and check their integrability. Moreover, we prove that in some cases
one can give K\"ahler and Nearly-K\"ahler metrics to these spaces.
