The curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature (at any point and time) and can be interpreted as the gradient flow of the length. We consider the same flow for networks (finite unions of sufficiently smooth curves whose end points meet at junctions). Because of the variational nature of the problem, one expects that for almost all the times the evolving network will possess only triple junctions where the unit tangent vectors forms angles of 120 degrees (regular junctions). However, even if the initial network has only regular junctions, this property is not preserved by the flow and junctions of four or more curves may appear during the evolution. The aim of this talk is first to describe the process of singularity formation and thento explain the resolution of such singularities and how to continue the flow in a classical PDE framework. This is a research in collaboration with Jorge Lira (Universidade Federal do Ceará), Rafe Mazzeo (Stanford University) and Mariel Sáez (Pontificia Universidad Católica de Chile).