states and the resulting rich structure of the low-energy effective field theories for these spin states ($SU(2)$, $U(1)$ and $Z_2$ spin liquids) are discussed. An interesting linkage of the $U(1)$ spin liquid state to the (metallic) Fermi liquid state through a Mott metal-insulator transition is introduced.

The difficulty of finding controllable approaches for studying spin liquid states has led to an extension of the RVB approach and a search for alternative approaches. Some of these approaches are reviewed briefly in section IV, including (i) the extension of the RVB approach to include the effect of spin-orbit coupling and to higher spin ($S>1/2$) systems, (ii) the concepts of matrix product states and projected entangled pair states, and (iii) the Kitaev honeycomb model. The main message of this section is that a larger variety of exotic spin states become possible when we leave the paradigm of spin $1/2$ systems with rotational symmetry. The $U(1)$ and $Z_2$ spin liquid states belong to merely a very small subset of the plausible exotic states once we leave the paradigm of semi-classical approaches.

Section V is devoted to a survey of experimental research on spin liquid states. Special attention is paid to the $U(1)$ spin liquid 