Single atoms and molecules can be trapped in tightly focused beams of light that form ‘optical tweezers’, affording exquisite capabilities for the control and detection of individual particles. This approach has progressed to creating tweezer arrays holding hundreds of atoms, resulting in a platform for controlling large many-particle quantum systems. Here we review this new approach to microscopic control of scalable atomic and molecular neutral quantum systems, its future prospects, and applications in quantum information processing, quantum simulation and metrology.Large arrays of atoms and molecules can be arranged and controlled with high precision using optical tweezers. This Review surveys the latest methodological advances and their applications to quantum technologies.