a locally compact group $g$ is \emph{hermitian} if the spectrum $\sigma_{l^1(g)}(f)$ is contained in $\mathbb r$ for every $f=f^*\in l^1(g)$. examples of hermitian groups include all abelian locally compact groups. a question from the 1960s asks whether every hermitian group is amenable. i will speak on the history and recent affirmative solution to this problem.