The geometry of the 2D Gaussian free field

The continuum Gaussian free field (GFF) can be seen as a generalization of Brownian motion to higher dimensions and it is a canonical example of a random height function. I will discuss some recent progress in describing geometric and probabilistic properties of the 2D GFF : How do its level sets look like? What is the structure of its excursions off the level sets? Answering these questions will reveal connections to SLE processes, to Brownian loop soups and even to the probabilistic theory of 2D Liouville quantum gravity.