The displacement of a single electron enables exciting avenues for nanotechnology with vast application potential in quantum metrology, quantum communication and quantum computation. Surface acoustic waves (SAW) have proven itself as a surprisingly useful solution to perform this task over large distance with outstanding precision and reliability. Over the last decade, important milestones have been achieved bringing SAW-driven single-electron transport from first proof-of-principle demonstrations to accurate, highly-controlled implementations, such as coherent spin transport, charge-to-photon conversion, or antibunching of charge states. Beyond the well-established piezoelectric gallium-arsenide platform, first realisations of acousto-electronic transport have also been carried out on the surface of liquid helium. In this review article, we aim to keep track of this remarkable progress by explaining these recent achievements from basic principles, with an outlook on follow-up experiments and near-term applications.