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The Gaussian free field (GFF) is a canonical model of random surfaces in probability theory, generalizing the Brownian bridge to higher dimensions. It arises naturally as the stationary solution to the stochastic heat equation with additive noise (SHE), and together the SHE and GFF are expected to be the universal scaling limit of many random surface evolution models arising in lattice statistical physics. We study the mixing time (time to converge to stationarity, when started out of equilibrium) for the central pre-limiting object, the discrete Gaussian free field (DGFF) evolving under the Glauber dynamics. In joint work with S. Ganguly, we establish that for every $d\ge 1$, on a box of side-length $n$ in $\mathbb Z^d$, the Glauber dynamics for the DGFF exhibits cutoff at time $\frac{d}{\pi^2} n^2 \log n$ with an $O(n^2)$ window. Our proof relies on an exact representation of the DGFF dynamics in terms of random walk trajectories with space-dependent jump times.