In this dissertation, we focus on the study of sharp inequalities of Moser- Trudinger-Onofri type. We first establish the analog Bliss and Hardy inequal- ities with sharp constant involving exponential weight function. One special case of the inequalities (for n = 2 ) leads to a direct proof of Onofri inequality on S2. Then we establish the sharp trace inequalities on any smooth bounded simply connected domain in R2.