In this talk, we will report some recent progress on sharp geometric and functional inequalities by using the Helgason-Fourier analysis techniques on hyperbolic and symmetric spaces. These techniques allow us to establish sharp higher order Hardy- Sobolev-Maz’ya and Hardy-Adams inequalities on upper half spaces, complex Siegel do- mains and quaternionic and octanionic hyperbolic spaces. Some applications to PDEs will also be given.