The article reviews the theory of one-dimensional (1D) Fermi liquids, focusing on the breakdown of Fermi liquid theory in 1D due to the Peierls divergence and charge-spin separation. The Luttinger model, an exactly solvable model, describes the low-energy properties of 1D quantum systems, incorporating these features. The model's eigenvalue spectrum is parameterized by a renormalized velocity and an effective coupling constant, fully describing its physics. Other 1D models share these properties in a low-energy subspace, and the concept of a "Luttinger liquid" describes their universality class. The Luttinger model provides an asymptotically exact solution for 1D many-body problems, applicable to lattice models like the 1D Hubbard model and systems with impurities or phonon interactions. The article discusses various solutions of the Luttinger model, emphasizing its physical properties, correlation functions, and the role of charge-spin separation. It also covers alternative methods, conformal field theory, and the behavior of systems outside the Luttinger liquid universality class. The review highlights the importance of the Luttinger liquid concept in understanding 1D fermions and its relevance to experimental observations in quasi-1D materials, charge density waves, and the quantum Hall effect. The article concludes with a summary of experimental evidence supporting Luttinger liquid correlations in various materials.The article reviews the theory of one-dimensional (1D) Fermi liquids, focusing on the breakdown of Fermi liquid theory in 1D due to the Peierls divergence and charge-spin separation. The Luttinger model, an exactly solvable model, describes the low-energy properties of 1D quantum systems, incorporating these features. The model's eigenvalue spectrum is parameterized by a renormalized velocity and an effective coupling constant, fully describing its physics. Other 1D models share these properties in a low-energy subspace, and the concept of a "Luttinger liquid" describes their universality class. The Luttinger model provides an asymptotically exact solution for 1D many-body problems, applicable to lattice models like the 1D Hubbard model and systems with impurities or phonon interactions. The article discusses various solutions of the Luttinger model, emphasizing its physical properties, correlation functions, and the role of charge-spin separation. It also covers alternative methods, conformal field theory, and the behavior of systems outside the Luttinger liquid universality class. The review highlights the importance of the Luttinger liquid concept in understanding 1D fermions and its relevance to experimental observations in quasi-1D materials, charge density waves, and the quantum Hall effect. The article concludes with a summary of experimental evidence supporting Luttinger liquid correlations in various materials.