The Gaussian free field is a random height function (technically a random distribution) that crops up frequently in both math and physics. In particular, it is the starting point for various constructions in Liouville quantum gravity. You’ve probably already encountered examples of the Gaussian free field: the Gaussian free field on $(0,\infty)$ is just standard Brownian motion and the Gaussian free field on $(0,1)$ is just the standard Brownian bridge. In this talk, we’ll define the Gaussian Free Field for arbitrary subdomains $U\in{\mathbb{R^{n}}}$ and look at some of its properties: i.e. the spatial Markov property, conformal invariance (in dimension 2), the behavior of circle averages and thick points.