Provided is an electric power steering system and a vehicle. The electric power steering system comprises a steering gear housing, a worm gear, and a worm. The steering gear housing is internally provided with a worm gear installation cavity and a worm installation cavity. The worm gear is installed in the worm gear installation cavity. The worm is installed in the worm installation cavity. The worm gear and the worm are meshed with each other. One axial end of the worm is provided with a first center distance adjusting mechanism, and the other axial end is provided with a second center distance adjusting mechanism. The first center distance adjusting mechanism and the second center distance adjusting mechanism can make the worm swing deviating the axis of the worm, so as to realize adjustment of the mesh center distance of the worm gear and the worm. Using the electric power steering system can realize mesh center distance of the worm gear and the worm, so that the mesh center distance of the worm gear and the worm is consistent with the center distance of the worm gear installation cavity and the worm installation cavity, and mesh interspace is not generated, so as to prevent to generate noise in a steering process.