This Thesis consists of two topics. The first topic (Chapter 2) is about extending the GLn x GLn -1 Rankin-Selberg integral to certain cases that the automorphic forms that are involved are relative rank one Eisenstein series. The main result is an expression of the residues as GLn-1 x GLn-2 Rankin-Selberg integrals for cusp forms. The other topic (Chapter 3) is about symplectic integrals of cusp forms on GL2n. The main result is a generalization of the Rankin-Selberg integral representing the exterior cube L--function on GL6. At this moment we cannot extract any information for L--functions from this identity. We hope that this will be possible in the future.
