  The Densest Subgraph Problem requires to find, in a given graph, a subset of
vertices whose induced subgraph maximizes a measure of density. The problem has
received a great deal of attention in the algorithmic literature since the
early 1970s, with many variants proposed and many applications built on top of
this basic definition. Recent years have witnessed a revival of research
interest in this problem with several important contributions, including some
groundbreaking results, published in 2022 and 2023. This survey provides a deep
overview of the fundamental results and an exhaustive coverage of the many
variants proposed in the literature, with a special attention to the most
recent results. The survey also presents a comprehensive overview of
applications and discusses some interesting open problems for this evergreen
research topic.
