In this dissertation, we consider three aspects: comparison theorems on complete manifold which posses a pole, geometric inequalities on complete manifolds, and the applications of inequalities to p-harmonic geometry. More precisely, wefirst derive a comparison theorem of the matrix-valued Riccati equation with certain initial conditions, and then use this as a tool to obtain

Hessian comparison theorem on manifolds with nonnegative curvatures. We study Hardy type inequality, weighted Hardy inequality and weighted Sobolev inequality via Hessian comparison theorems. One of the main results in this dissertation is the Caffarelli-Kohn-Nirenberg type inequality on Cartan- Hadamard manifolds, which is an extension of the the result in Caffarelli-

Kohn-Nirenberg's paper [6]. Furthermore, we also discuss some Lp version of Caffarelli-Kohn-Nirenberg type inequalities on punched manifolds and point out a possible value of the constant. Finally, we study Liouville theorems of p-harmonic functions, p-harmonic morphisms, and weakly conformal maps, with assumption only on curvature and q-energy growth. As further applications we obtain Picard type theorems in p-harmonic geometry.