"Recent Progress in the Random Conductance Model" Martin Barlow, University of British Columbia Consider the standard Euclidean lattice, and put random i.i.d. 'conductances' V(e) on each bond. We allow the possibility that V(e) is zero. Let Y be a continuous time Markov chain which jumps along the edge e with probability proportional to V(e). We assume that the probability that V(e) is positive is greater than p_c, the critical probability for bond percolation on the Euclidean lattice. Thus there exists (a.s.) a unique infinite connected subgraph on which Y can run. A special case of the above is when V(e) is either 0 or 1, and so Y is a random walk on a supercritical percolation cluster. Various kinds of 'trapping' can arise if V(e) can take either small positive values, or large values. In this talk I will discuss invariance principles for Y, and Gaussian bounds for its transition densities.