Rapidly propagating cracks are known to undergo a variety of dynamic instabilities. While some of these bear similarities to other well-understood instabilities in condensed matter physics, e.g. side-branching in dendritic crystal growth, as of yet we have no comparable understanding of dynamic fracture instabilities. In this talk I will briefly review the experimental observation of various crack instabilities and explain the origin of the difficulty. I will then discuss recent progress in understanding some of these instabilities: the “Weakly nonlinear theory of dynamic fracture” and its relation to the oscillatory crack instability in 2D and a crack front corrugation instability in 3D. Open challenges will be briefly discussed.