The quantum transport phenomena observed in semiconductor-based nanostructures over the past decade are described by the leading researchers in the field. Key articles describe the status of the quantum Hall effect, quantum dot transport, the theory of mesoscopic transport properties and a survey of mesoscopic transport phenomena, incompressible electron fluids, the Luttinger liquid, quantum chaology, quantum coherence and interactions, spectroscopy of nanostructures, and Cooper pair tunnelling. The most recent developments are described in a number of shorter contributions, providing a survey of the present status of research in this extremely active area of fundamental and applied condensed matter research. The presentation is suitable for advanced students and Ph.D. researchers wishing to gain an insight into the most recent work in the area.