Conventional optical tweezers are generated by the intensity gradient of highly focused laser beams, but the requirement of strong intensity gradient limits the tunability of optical traps. Here we show a new type of optical tweezers with tunable potential wells by manipulating the phase gradient of light. Using a new method to calculate holograms, we can design desirable phase profiles and intensity distributions of optical patterns. Optical force arising from the phase gradient creates tunable potential wells for versatile optical manipulation, such as trapping nanoparticles in peanut-shaped optical spots, and positioning and shifting nanoparticles in optical gears as demonstrated in our experiments. The phase-gradient-based optical tweezers have several merits including flexible design and easy control, which open a new way for optical trapping and manipulation.