I will begin by reviewing the general relations that exist between the cover time of graphs by random walk and the Gaussian free field on the graph, and explain the strength and limitations of these general methods. I will then discuss recent results concerning the cover time of the binary tree of depth $n$ by simple random walk, and in particular sharp fluctuation results for the cover time, mirroring those for the maximal displacement of branching random walk; certain barrier estimates for Bessel processes play a crucial role. Finally, I will describe how these technique can be applied to the study of the cover time of the 2-sphere by the Brownian sausage.