The curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature, and can be interpreted as the gradient flow of the length. In this talk I will consider the same flow for networks (finite unions of sufficiently smooth curves whose endpoints meet at junctions). I will explain how to define the flow in a classical PDE framework, and then I will list some examples of singularity formation, both at finite and infinite time, and explain the resolution of such singularities obtained by geometric microlocal analysis techniques. I will describe a stability result based on Lojasiewicz–Simon gradient inequalities and give a rough estimate on the basin of attraction of critical points. Furthermore, I will motivate the coarsening-type behavior clearly visible in numerical simulations. This seminar is mainly based on recent papers in collaboration with Jorge Lira (Uni- versidade Federal do Ceará), Rafe Mazzeo (Stanford University), Mariel Saez (P. Universidad Catolica de Chile) and Marco Pozzetta (Università di Napoli Federico II).