We study diffusion (random walks) on recursive scale-free graphs, and contrast the results to similar studies in other analytically soluble media. This allows us to identify ways in which diffusion in scale-free graphs is special. Most notably, scale-free architecture results in a faster transit time between existing nodes, when the network grows in size; and walks emanating from the most connected nodes are recurrent, despite the network's infinite dimension. We also find that other attributes of the graph, besides its scale-free distribution, have a strong influence on the nature of diffusion.