Quantum spin liquids (QSLs) and unconventional superconductivity (u-SC) are central topics in condensed matter physics, and their possible relationship has attracted considerable attention. In underdoped cuprates, u-SC is often discussed in connection with QSLs, in which magnetic excitations form spinons. However, the parent Mott insulators in cuprates are typically antiferromagnetically ordered, and competing or coexisting spin/charge orders complicate the relationship. In contrast, organic conductors κ-(BEDT-TTF)₂X with anisotropic triangular lattices provide a cleaner platform, including the Mott-insulating QSL κ-(BEDT-TTF)₂Cu₂(CN)₃ and its doped counterpart κ-(BEDT-TTF)₄Hg₂.₈₉Br₈, which becomes superconducting at T_c ≈ 4 K. I first discuss the QSL nature of κ-Cu₂(CN)₃; magnetization, NMR, ESR, and μSR results are consistent with a Z₂ nodal QSL at low temperatures, possibly crossing over to a U(1) spinon-Fermi-surface-like state at elevated temperatures. I then present quantum-critical non-Fermi-liquid and u-SC behaviors of κ-HgBr. In particular, I argue that magnetically pseudogapped behavior, reduced superfluid density, large T_c/T_F (~0.1), BEC-like character of Cooper pairs despite d-wave pairing, and unusually large upper critical fields are difficult to reconcile with the BCS framework, but rather appear consistent with a picture of fractionalized superconductivity, in which spinon pairing and holon BEC occur separately.