We present an experimental study of the rheology of polydisperse aqueous foams of different gas volume fractions phi. With oscillatory deformation at fixed frequency, we determine the behavior of the maximum stress as a function of the strain amplitude. At low strain, the maximum stress increases linearly, defining a shear modulus G. At progressively higher strains, the response eventually becomes nonlinear, defining the yield strain and the yield stress. While phi decreases toward phi(c) = 0.635+/-0.01, G goes to zero, and the yield stress decreases by many orders of magnitude with a quadratic behavior. The yield strain, which can be extrapolated to 0.18+/-0.02 at phi = 1, has a minimum value of 0.045+/-0.010 at phi(c). This behavior shows the occurrence of a melting transition located at phi(c), which can be correlated to the random close packing of spheres. We compare these results to similar ones obtained previously for monodisperse and polydisperse emulsions. Our new experiments clarify the rheological similarities between emulsions and foams, as well as the role of polydispersity. We find that as long as polydispersity is moderate, it does not play a crucial role in the elastic response of foams and emulsions.