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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.

New curvature flows in complex geometry