The Gaussian free field and the Liouville measure

The Gaussian free field (GFF) is  the generalisation of Brownian motion when the time set $R^+$ is replaced with a $d$-dimensional domain. It has been a center piece in the study of conformal invariance random geometry in the last 20 years. In this mini-course, we will construct and study the Gaussian free field, mostly in $d=2$. We will study its basic geometric properties, its generalised Markov property and finally, we will construct its exponential: The Liouville measure.