Robotic manipulation of small objects has shown great potential for engineering, biology, and chemistry research. However, existing robotic platforms have difficulty in achieving contactless, high-resolution, 4-degrees-of-freedom (4-DOF) manipulation of small objects, and noninvasive maneuvering of objects in regions shielded by tissue and bone barriers. Here, we present chirality-tunable acoustic vortex tweezers that can tune acoustic vortex chirality, transmit through biological barriers, trap single micro- to millimeter-sized objects, and control object rotation. Assisted by programmable robots, our acoustic systems further enable contactless, high-resolution translation of single objects. Our systems were demonstrated by tuning acoustic vortex chirality, controlling object rotation, and translating objects along arbitrary-shaped paths. Moreover, we used our systems to trap single objects in regions with tissue and skull barriers and translate an object inside a Y-shaped channel of a thick biomimetic phantom. In addition, we showed the function of ultrasound imaging-assisted acoustic manipulation by monitoring acoustic object manipulation via live ultrasound imaging.