Abstract: In 1988 Adams used Riesz potentials as a tool to prove the higher order  version of Moser-Trudinger sharp exponential inequality. Since potential-like operators are very flexible and cover  several examples of growing importance, fractional powers of elliptic operators to mention one, we believe that sharp exponential inequalities for these operators are interesting  in themselves. In the talk we will present theorems that explore their behaviour at the Sobolev critical index, and not only, in unbounded domains of `\mathbb{R}^n`.