We review the interplay of frustration and strong electronic correlations in quasi-two-dimensional organic charge transfer salts, such as (BEDT-TTF)2X and EtnMe4−nPn[Pd(dmit)2]2. These two forces drive a range of exotic phases including spin liquids, valence bond crystals, pseudogapped metals and unconventional superconductivity. Of particular interest is that in several materials pressure drives a first-order transition from a spin liquid Mott insulating state to a superconducting state. Experiments on these materials raise a number of profound questions about the quantum behaviour of frustrated systems, particularly the intimate connection between spin liquids and superconductivity. Insights into these questions have come from a wide range of theoretical techniques including first principles electronic structure, quantum many-body theory and quantum field theory. In this review we introduce some of the basic ideas of the field by discussing a simple frustrated Heisenberg model with four spins. We then describe the key experimental results, emphasizing that for two materials, κ-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2, there is strong evidence for a spin liquid ground state, and for another, EtMe3P[Pd(dmit)2]2, there is evidence of a valence bond crystal ground state. We review theoretical attempts to explain these phenomena, arguing that they can be captured by a Hubbard model on the anisotropic triangular lattice at half filling, and that resonating valence bond wavefunctions capture most of the essential physics. We review evidence that this Hubbard model can have a spin liquid ground state for a range of parameters that are realistic for the relevant materials. In particular, spatial anisotropy and ring exchange are key to destabilizing magnetic order. We conclude by summarizing the progress made thus far and identifying some of the key questions still to be answered.