Cohomology jump loci are homotopy invariants associated to topological spaces of finite homotopy type. They are generalizations of usual cohomology groups. I will give a survey on the theory of cohomology jump loci of projective, quasi-projective and compact Kahler manifolds, due to Carlos Simpson, Nero Budur and myself. In the second part of the talk, I will introduce some concrete examples of 6-dimensional symplectic-complex Calabi-Yau manifold, which satisfies all the known topological criterions of compact Kahler manifolds such as Hodge theory and Hard Lefschetz theorem, but fail the cohomology jump locus property of compact Kahler manifolds. The second part is joint work with Lizhen Qin.