  The present survey is devoted to results on Trudinger-Moser inequalities in
two dimension. We give a brief overview of the history of these celebrated
inequalities and, starting from the geometric problem that motivated Moser's
original work, we discuss the connection between Onofri's inequality for the
unit sphere and sharp inequalities on Euclidean domains. Finally, we present
recent results and new insights into nonlocal interaction energy functionals in
two dimension, involving logarithmic kernels.
