A milestone for QED is the attainment of the Schwinger field, an electric field strong enough to induce e+e- pair creation, E_cr = m^2/e in natural units. With recent progress in high-power lasers, it is now possible to build experiments in which an electron from an accelerator sees the Schwinger field in its rest frame. Experiments in this regime are now planned at SLAC and DESY. In this talk, I will discuss the physics of such electron-laser beam collisions. What are the predictions of QED for crossing E ~ E_cr? What do we still need to learn about QED in this regime ? What applications need this knowledge?