This article provides an introductory review of the physics of quantum spin liquid (QSL) states. Quantum magnetism is a rapidly evolving field, and recent developments reveal that the ground states and low-energy physics of frustrated spin systems may exhibit many exotic behaviors once we move beyond semi-classical approaches. The purpose of this article is to introduce these developments. It begins by explaining how semi-classical approaches fail when quantum mechanics becomes important and describes alternative approaches for addressing the problem. The article focuses mainly on spin 1/2 systems and discusses one particular set of plausible spin liquid states, which are spin-singlet states and can be viewed as an extension of Fermi liquid states to Mott insulators. These states are usually classified as SU(2), U(1), or Z2 spin liquid states. The article reviews the basic theory of these states and their extensions to include spin-orbit coupling and higher spin systems. It also introduces two other important approaches: matrix product states and projected entangled pair states, and the Kitaev honeycomb model. Experimental progress on spin liquid states in real materials, including anisotropic triangular lattice systems and kagome lattice systems, is reviewed and compared with corresponding theories. The article is organized into sections discussing semi-classical to non-linear-σ model approaches, resonant valence bond (RVB) states, and QSL states in real materials. The article concludes with a summary of the key findings and discusses the implications of these results for the understanding of quantum spin liquids.This article provides an introductory review of the physics of quantum spin liquid (QSL) states. Quantum magnetism is a rapidly evolving field, and recent developments reveal that the ground states and low-energy physics of frustrated spin systems may exhibit many exotic behaviors once we move beyond semi-classical approaches. The purpose of this article is to introduce these developments. It begins by explaining how semi-classical approaches fail when quantum mechanics becomes important and describes alternative approaches for addressing the problem. The article focuses mainly on spin 1/2 systems and discusses one particular set of plausible spin liquid states, which are spin-singlet states and can be viewed as an extension of Fermi liquid states to Mott insulators. These states are usually classified as SU(2), U(1), or Z2 spin liquid states. The article reviews the basic theory of these states and their extensions to include spin-orbit coupling and higher spin systems. It also introduces two other important approaches: matrix product states and projected entangled pair states, and the Kitaev honeycomb model. Experimental progress on spin liquid states in real materials, including anisotropic triangular lattice systems and kagome lattice systems, is reviewed and compared with corresponding theories. The article is organized into sections discussing semi-classical to non-linear-σ model approaches, resonant valence bond (RVB) states, and QSL states in real materials. The article concludes with a summary of the key findings and discusses the implications of these results for the understanding of quantum spin liquids.