  This article is 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 develop many exotic behaviors once we leave the regime of
semi-classical approaches. The purpose of this article is to introduce these
developments. The article begins by explaining how semi-classical approaches
fail once quantum mechanics become important and then describes the alternative
approaches for addressing the problem. We discuss mainly spin $1/2$ systems,
and we spend most of our time in this article on one particular set of
plausible spin liquid states in which spins are represented by fermions. These
states are spin-singlet states and may be viewed as an extension of Fermi
liquid states to Mott insulators, and they are usually classified in the
category of so-called $SU(2)$, $U(1)$ or $Z_2$ spin liquid states. We review
the basic theory regarding these states and the extensions of these states to
include the effect of spin-orbit coupling and to higher spin ($S>1/2$) systems.
Two other important approaches with strong influences on the understanding of
spin liquid states are also introduced: (i) matrix product states and projected
entangled pair states and (ii) the Kitaev honeycomb model. Experimental
progress concerning spin liquid states in realistic materials, including
anisotropic triangular lattice systems ($\kappa$-(ET)$_{2}$Cu$_{2}$(CN)$_{3}$
and EtMe$_{3}$Sb[(Pd(dmit)$_{2}$]$_{2}$), kagome lattice systems
(ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$) and hyperkagome lattice systems
(Na$_{4}$Ir$_{3}$O$_{8}$), is reviewed and compared against the corresponding
theories.
