  In recent years, log-concave density estimation via maximum likelihood
estimation has emerged as a fascinating alternative to traditional
nonparametric smoothing techniques, such as kernel density estimation, which
require the choice of one or more bandwidths. The purpose of this article is to
describe some of the properties of the class of log-concave densities on
$\mathbb{R}^d$ which make it so attractive from a statistical perspective, and
to outline the latest methodological, theoretical and computational advances in
the area.
