Cohomology jump loci are generalizations of usual cohomology groups of a topological space. In the first part of the talk, I will give a survey on the recent development of the theory of cohomology jump loci of complex algebraic varieties. In the second part of the talk, I will use some concrete examples of (real) 6-dimensional symplectic-complex Calabi-Yau manifolds to illustrate how cohomology jump loci can give new constraints on the topology of compact Kahler manifolds.