Many fundamental processes in physics involve transport. For a variety of physical systems, e.g. electrons, light, cold atoms and sound, transport mechanisms eventually reduce to different manifestations of wave transport. In the last decades, pushed by the spectacular progresses in the control and engineering of matter at the nano-scale, new regimes of wave transport became of strong interest. Indeed, fascinating effects emerge when transport is studied at the “nano” level, when atoms behave like waves and light propagation in nano-engineered structures acquires intriguing behaviors. This book collects contributions from speakers and lecturers of the CLXXIII International School of Physics “Enrico Fermi” which was held in Varenna (Italy) from June 23rd to July 3rd 2009. Different aspects of wave transport were covered during the school, from electrons to light propagation, from sound to ultracold atoms. Considering the ubiquitous nature of wave transport phenomena, the school was characterized by a strongly interdisciplinary approach, with speakers, lecturers and students from different communities meeting and sharing their knowledge and the often complementary points of view and approaches.

Among the different media in which waves can travel, periodic and disordered structures surely deserve particular attention. Interference of waves in periodic structures results in the formation of energy bands, which are responsible for the conduction properties of electrons in solids. Periodic structures can be realized also for light and ultra-cold atoms, in the form of photonic crystals or optical lattices, respectively, which allow the observation of effects which have been originally predicted in the context of solid-state physics. The most recent advances in the physics of ultra-cold atoms in optical lattices are discussed in the contribution by I. Bloch concerning quantum simulation of condensed-matter physics. Optical lattices also allow the production of low-dimensional atomic systems, as discussed in the paper by Z. Hadzibabic and J. Dalibard devoted to the investigation of transport and superfluidity in 2D bosonic quantum gases.

Disordered structures also show fascinating phenomena. Multiple scattering in random media results in the localization of waves predicted by P. W. Anderson fifty years ago for electrons moving in disordered crystals, and reviewed in this book in the opening article by P. Woefle. Also in this case, a phenomenon originally predicted for electrons in crystals has been observed both for propagation of classical waves — light and sound — in disordered media and very recently for ultracold atoms expanding in disordered optical potentials. Transport of sound waves in different media, including Anderson localization in disordered structures, is discussed in the contributions by J. Page.

At the border between periodic and disordered media, quasicrystals are topological structures showing long-range order and absence of periodicity, which results in intriguing properties that are described in the contribution by P. J. Steinhardt. Optical quasicrystals can be realized for ultra-cold atoms and used to study Anderson localization of matter waves, as discussed in the article by L. Fallani and M. Inguscio.

Knowledge of transport properties in complex systems is important not only for fundamental studies, but also for applications. Understanding the propagation of light is extremely important for engineering new devices, as metamaterials and plasmonic materials, and for applications in the field of energy, biology and medicine. In this perspective, the article by R. C. Mesquita and A. G. Yodh covers the application of diffuse optics to medical imaging. Extending these concepts beyond the field of optics, in two different contributions to this book, M. Fink and coworkers discuss multi-wave imaging for medical applications and present theory and applications of time-reversal focusing.

The success of the Summer School was not only determined by the high quality of the lectures, but also by the enthusiasm of the students and observers who attended the Course. Their active participation resulted in the success of the two poster sessions (the most interesting posters have been upgraded to invited presentations) and of the final discussion session on future research perspectives. We would like to warmly thank all the speakers, lecturers, participants and express our gratitude to the organizing team, in particular Barbara Alzani of the Italian Physical Society for her passion and dedication in the Course organization, as well as Ramona Brigatti and Marta Pigazzini for their enthusiasm and assistance in Varenna. We also acknowledge financial support from the Italian Physical Society through its president Luisa Cifarelli, and from the European network Intercan.

Finally, the Summer School hosted a celebration in memoriam of Franco Bassani (1929-2008), former president of the Italian Physical Society. On this occasion, friends and colleagues Lucio Andreani, Luisa Cifarelli, Massimo Inguscio and Erio Tosatti presented several portraits of his scientific and personal life. Franco Bassani was an excellent scientist and an outstanding man. We dedicate this book to his memory.

R. Kaiser, D. S. Wiersma and L. Fallani