Interactions among quantum particles lead to remarkable phenomena. For fermions a cornerstone of our understanding of such effects is the Fermi liquid theory, due to Landau.

However, in certain situations such as one and quasi-one dimensional systems, the Fermi liquid theory fails and interactions lead to a radically new set of properties, generically described under the name of Luttinger liquids.

These concepts will be reviewed and the formalism and the physics both for bosonic and fermionic systems will be described in details. Applications to condensed matter systems and also to the new phases that could be obtained in ultracold atomic gases will be tackled.