Abstract: We discuss the holonomy of Riemannian manifolds and geometries, which arise from the irreducible holonomy groups with an emphasis on Calabi-Yau manifolds and hyperkahler manifods (the case of SU(n) and Sp(n) holonomy respectfully). I will prove some structural results about the Calabi-Yau manifolds and give some examples in compact and non-compact cases. If time permits, we will discuss calibrated submanifolds (i.e. complex and special Lagrangian) in Calabi-Yau manifolds and their deformations.