Quivers are directed graphs which encode information about the gauge groups and matter content of a large class of gauge theories, many of which have AdS/CFT duals. The counting of local gauge invariant operators and the computation of their correlators (in the free field limit) can be done by simple diagrammatic manipulations of the quiver, with the help of permutation group theory data. This data includes Young diagrams, Littlewood-Richardson numbers and branching coefficients of permutation groups. Riemann surfaces obtained by thickening the quivers are intimately related to these computations.