Abstract: Sharp geometric inequalities play an important role in analysis, PDEs and differential

geometry. In this talk, I will review some classical Hardy, Hardy-Sobolev and Hardy-Sobolev-

Maz'ya inequalities of first order and describe some recent works on the higher order Hardy-

Sobolev-Maz'ya and Hardy-Adams inequalities on hyperbolic balls and half spaces.  The

relationship between the classical Sobolev inequalities and the Hardy-Sobolev-Maz'ya inequalities

for higher order derivatives will be discussed. If time allows, we then briefly describe how the techniques of Fourier analysis on hyperbolic spaces and Green's function estimates can be used to establish such inequalities.