We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In particular, we give a complete proof of global existence and smooth convergence to critical points of the solution of the elastic flow of closed curves in $\mathbb{ R}^2$. In the last section of the paper we also discuss a list of open problems.
Categories: math.AP math.DG