The Kirkendall effect is a consequence of the different diffusivities of atoms in a diffusion couple causing a supersaturation of lattice vacancies. This supersaturation may lead to a condensation of extra vacancies in the form of so-called "Kirkendall voids" close to the interface. On the macroscopic and micrometer scale these Kirkendall voids are generally considered as a nuisance because they deteriorate the properties of the interface. In contrast, in the nanoworld the Kirkendall effect has been positively used as a new fabrication route to designed hollow nano-objects. In this Review we summarize and discuss the demonstrated examples of hollow nanoparticles and nanotubes induced by the Kirkendall effect. Merits of this route are compared with other general methods for nanotube fabrication. Theories of the kinetics and thermodynamics are also reviewed and evaluated in terms of their relevance to experiments. Moreover, nanotube fabrication by solid-state reactions and non-Kirkendall type diffusion processes are covered.