The Kondo effect is relevant to many branches of condensed matter physics, including heavy-fermion physics, correlated electron systems, and high-temperature superconductivity. Nanolithographic fabrication techniques and new spectroscopic measuring tools (scanning tunnelling microscopy, mechanically controllable break junctions, for example) mean we can probe the Kondo effect directly on the relevant mesoscopic size scale. This book brings together the major experimental and theoretical progress that has been made on low-dimensional Kondo systems. The following questions are addressed: the relevance of the Kondo screening cloud, surface-induced anisotropy and disorder to an understanding of the size dependence of the Kondo transport properties in small metallic samples of dilute magnetic alloys; what measurements with scanning tunnelling spectroscopy and with mechanically controllable break junctions teach us about the spatial extent of the Kondo effect and the relevance of local fluctuations in the electronic density of states; the relevance of two-level systems for electron transport through metallic nanobridges, and how the interaction with two-level systems can be described in terms of two-channel Kondo scattering; is the experimentally observed saturation at low temperatures of the electron dephasing time an intrinsic effect or is it due to the presence of two-level systems or residual magnetic impurities?; and how can quantum dots be used as perfectly tunable Kondo systems?