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The coupling between homological cycles and differential forms representing cohomological classes appears in a classical setting of the topological study of an algebraic variety. We consider a family of complex algebraic varieties depending on deformation parameters and study the above mentioned coupling as ramifying functions (known as period integrals) depending on deformation parameter variables. This ramification (monodromy) happens around the discriminantal loci of the deformation family. The monodromy data of the period integrals are equivalent to those of vanishing homological cycles. In this way we get an analytic description of the monodromy of the cycles (topological in nature) through period integrals (analytic objects). After our method, in some important cases (e.g. affine toric variety), the period integral can be interpreted as a generalized hypergeometric function. In combining results known before (e.g. GKZ, Horn HGF) and newly invented ones we shall study the monodromy of the period integrals in global settings (monodromy group) as well as in local settings. Exact knowledge on the monodromy group of the period integrals allows us to establish results related to homological mirror symmetry conjecture or to Dubrovin’s conjecture on the Stokes matrix of the quantum cohomology of Fano variety. We use mainly the method of analytic continuation to establish these results.