We will discuss some recent results in the direction of sharp constants for the Moser-Trudinger inequality and Adams inequality in high order Sobolev spaces.We develop a new method to prove such inequalities in unbounded domains without using symmetrization.This argument is substantially different from those in the literature where symmetrization was crucial.It enables us to derive Adams inequalities in high order Sobolev spaces of arbitrary orders and thus prove such theorems in full generality.It can also be used to obtain sharp Moser-Trudinger inequalities in other more general settings including the Heisenberg group where symmetrization is not available.This is joint work with Nguyen Lam.