  This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or
Onofri inequality for brevity. In dimension two this inequality plays a role
similar to the Sobolev inequality in higher dimensions. After justifying this
statement by recovering the Onofri inequality through various limiting
procedures and after reviewing some known results, we state several elementary
remarks. We also prove various new results. We give a proof of the inequality
using mass transportation methods (in the radial case), consistently with
similar results for Sobolev's inequalities. We investigate how duality can be
used to improve the Onofri inequality, in connection with the logarithmic
Hardy-Littlewood-Sobolev inequality. In the framework of fast diffusion
equations, we establish that the inequality is an entropy--entropy production
inequality, which provides an integral remainder term. Finally we give a proof
of the inequality based on rigidity methods and introduce a related nonlinear
flow.
