We report wafer-scale growth of atomically thin, three-dimensional (3D) van der Waals (vdW) semiconductor membranes. By controlling the growth kinetics in the near-equilibrium limit during metal-organic chemical vapor depositions of MoS2 and WS2 monolayer (ML) crystals, we have achieved conformal ML coverage on diverse 3D texture substrates, such as periodic arrays of nanoscale needles and trenches on quartz and SiO2/Si substrates. The ML semiconductor properties, such as channel resistivity and photoluminescence, are verified to be seamlessly uniform over the 3D textures and are scalable to wafer scale. In addition, we demonstrated that these 3D films can be easily delaminated from the growth substrates to form suspended 3D semiconductor membranes. Our work suggests that vdW ML semiconductor films can be useful platforms for patchable membrane electronics with atomic precision, yet large areas, on arbitrary substrates.