state, on which most experimental efforts have been focused. The best studied examples are a family of organic compounds denoted by $\kappa$-(ET)$_{2}$Cu$_{2}$(CN)$_{3}$ (ET) [@Kanoda03] and Pd(dmit)$_{2}$(EtMe$_{3}$Sb) (dmit salts) [@Itou08]. Both materials are Mott insulators near the metal-insulator transition and become superconducting (ET) or metallic (dmit) under modest pressure. Despite the large magnetic exchange $J\approx 250$ K observed in these systems, there is no experimental indication of long-range magnetic ordering down to a temperature of $\sim30$ mK. A linear temperature dependence of the specific heat and a Pauli-like spin susceptibility have been found in both materials at low temperature, suggesting that the low-energy excitations are spin-$1/2$ fermions with a Fermi surface [@SYamashita; @Matsuda12]. This Fermi-liquid-like behavior is further supported by their Wilson ratios, which are close to one. In addition to ET and dmit salts, the kagome compound ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ [@Kagome07] and the three-dimensional hyperkagome material Na$_{4}$Ir$_{3}$O$_{8}$ [@Na4Ir3O8] are also considered to be candidates for QSLs with gapless excitations. Experimental surveys on these QSL candidate materials are presented in this article, including their thermodynamics, thermal transport and various spin spectra. We also briefly introduce the discoveries of a few new materials and discuss the existing discrepancies between experiments and theories. The paper 