This article reviews the progress in the theory of one-dimensional (1D) Fermi liquids over the past decade. The traditional Fermi liquid theory, based on a quasi-particle picture, breaks down in 1D due to the Peierls divergence in the particle-hole bubble and charge-spin separation. These issues are related to the importance of scattering processes transferring finite momentum. The Luttinger model, an exactly solvable model that incorporates these features, provides a framework for describing the low-energy properties of gapless 1D quantum systems. The model's correlation functions can be calculated, and its eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describes the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, which constitutes the universality class of these quantum systems. Once the mapping to the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models such as the 1D Hubbard model off half-filling, variants like the $t-J$ or extended Hubbard model, and 1D electron-phonon systems or metals with impurities can be Luttinger liquids. The article discusses various solutions of the Luttinger model, emphasizing different aspects of the physics of 1D Fermi liquids. It also explores the generic behavior of systems not falling into the Luttinger liquid universality class due to gaps in their low-energy spectrum, such as the Mott transition. The article concludes with a summary of experiments that provide evidence for Luttinger liquid correlations in the "normal" state of quasi-1D organic conductors, superconductors, charge density wave systems, and semiconductors in the quantum Hall regime.This article reviews the progress in the theory of one-dimensional (1D) Fermi liquids over the past decade. The traditional Fermi liquid theory, based on a quasi-particle picture, breaks down in 1D due to the Peierls divergence in the particle-hole bubble and charge-spin separation. These issues are related to the importance of scattering processes transferring finite momentum. The Luttinger model, an exactly solvable model that incorporates these features, provides a framework for describing the low-energy properties of gapless 1D quantum systems. The model's correlation functions can be calculated, and its eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describes the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, which constitutes the universality class of these quantum systems. Once the mapping to the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models such as the 1D Hubbard model off half-filling, variants like the $t-J$ or extended Hubbard model, and 1D electron-phonon systems or metals with impurities can be Luttinger liquids. The article discusses various solutions of the Luttinger model, emphasizing different aspects of the physics of 1D Fermi liquids. It also explores the generic behavior of systems not falling into the Luttinger liquid universality class due to gaps in their low-energy spectrum, such as the Mott transition. The article concludes with a summary of experiments that provide evidence for Luttinger liquid correlations in the "normal" state of quasi-1D organic conductors, superconductors, charge density wave systems, and semiconductors in the quantum Hall regime.