Moser-Trudinger inequalities arise naturally in the study of the critical case of the well known Sobolev embeddings. In this work, we cover two issues related to Moser-Trudinger inequalities. We address the problem of the existence of extremal functions for Moser-Trudinger embeddings in the presence of singular potentials, and the problem of finding sharp Moser-Trudinger type inequalities in fractional Sobolev spaces
