The three-dimensional particle-in-cell (PIC) code VLPL (Virtual Laser Plasma Lab) allows, for the first time, direct fully electromagnetic simulations of relativistic laser–plasma interactions. Physical results on relativistic self-focusing in under-dense plasma are presented. It is shown that background plasma electrons are accelerated to multi-MeV energies and 104 T magnetic fields are generated in the process of self-focusing at high laser intensities. This physics is crucial for the fast ignitor concept in inertial confinement fusion. Advances in the numerical PIC algorithm used in the code VLPL are reviewed here.