We report the results of both angle-dispersive x-ray diffraction and x-ray absorption near-edge structure studies in BaWO4 and PbWO4 at pressures of up to 56 GPa and 24 GPa, respectively. BaWO4 is found to undergo a pressure-driven phase transition at 7.1 GPa from the tetragonal scheelite structure (which is stable under normal conditions) to the monoclinic fergusonite structure whereas the same transition takes place in PbWO4 at 9 GPa. We observe a second transition to another monoclinic structure which we identify as that of the isostructural phases BaWO4-II and PbWO4-III (space group P21/n). We have also performed ab initio total energy calculations which support the stability of this structure at high pressures in both compounds. The theoretical calculations further find that upon increase of pressure the scheelite phases become locally unstable and transform displacively into the fergusonite structure. The fergusonite structure is however metastable and can only occur if the transition to the P21/n phases were kinetically inhibited. Our experiments in BaWO4 indicate that it becomes amorphous beyond 47 GPa. Comment: 46 pages, 11 figures, 3 tables