Waves with nonzero angular momentum have shown capability in boosting communication efficiency, particle manipulation, and creating source illusions, among many other possibilities. Generating such waves has led inevitably to scattering, which limits their performance, and nearly perfect generation of high-order angular momentum remains challenging. This study presents experimental realization of theoretically perfect creation of wavefronts with angular momentum via metasurfaces. This design strategy shows the way to practical, highly efficient metasurfaces for various functionalities, and to complete control of fields radiated by compact sources.