  This is a survey of some of the recent developments on the geometric and
analytic aspects of the Anomaly flow. It is a flow of $(2,2)$-forms on a
$3$-fold which was originally motivated by string theory and the need to
preserve the conformally balanced property of a Hermitian metric in the absence
of a $\partial\bar\partial$-Lemma. It has revealed itself since to be a
remarkable higher order extension of the Ricci flow. It has also led to several
other curvature flows which may be interesting from the point of view of both
non-K\"ahler geometry and the theory of non-linear partial differential
equations.
