The electronic structure of PbWO4 crystal is studied with the relativistic self-consistent discrete variational embedded cluster method in the framework of density functional theory. The results show that the valence band is mainly made up of O2p with a certain amount of W5d component, and the conduction band is mainly made up of the hybrid states of W5d and O2p. The bottom of the conduction band is first found to be occupied by a narrow Pb6p1/2 state. The band gap and the width of valence band are around 4.8 and 4.0eV, respectively. The results can successfully interpret the reflective spectrum of the crystal. The mechanism of the blue luminescence of the crystal is discussed.