Sharp geometric inequalities play important roles in analysis, geometry, mathematical physics, PDEs and many other branches of modern mathematics. In this talk, we will report progress on using the techniques of Helgason-Fourier analysis on hyperbolic spaces to establish sharp gemetric and functional inequalities we have developed in recent years. These include higher order Poincare and Sobolev inequalities and Hardy-Sobolev-Maz'ya inequalities for GJMS operators on hyperbolic spaces and their best constants, Hardy- Moser-Trudinger and Hardy-Adams inequalities and their best constants, etc. The Helgason-Fourier analysis techniques play a crucial role in our approaches different from tools in the literature.