Abstract: We outline recent work towards a global well-posedness theory for the massless cubic Dirac equation for small, scale invariant data in spatial dimensions n = 2, 3. The main difficulty is the lack of available Strichartz estimates for the Dirac equation in low dimensions. To overcome this, there are two main steps. The first is a construction of the null frame spaces of Tataru that is adapted to the Dirac equation, and which form a suitable replacement for certain missing endpoint Strichartz estimates. The second is a number of bilinear and trilinear estimates that exploit the null structure of the equation.