  In recent work, Gr\"unrock and Pecher proved that the Dirac-Klein-Gordon
system in 2d is globally well-posed in the charge class (data in $L^2$ for the
spinor and in a suitable Sobolev space for the scalar field). Here we obtain
the analogous result for the full Maxwell-Dirac system in 2d. Making use of the
null structure of the system, found in earlier joint work with Damiano Foschi,
we first prove local well-posedness in the charge class. To extend the
solutions globally we build on an idea due to Colliander, Holmer and Tzirakis.
For this we rely on the fact that MD is charge subcritical in two space
dimensions, and make use of the null structure of the Maxwell part.
