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Let $U \subseteq \C$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\int_U \nabla f(x) \cdot \nabla g(x) dx$. The set $T_U(a)$ of $a$-thick points of $F$ consists of those $z \in U$ such that the average of $F$ on a disk of radius $r $ centered at $z$ has growth $\sqrt{a/\pi} \log \tfrac{1}{r}$ as $r \to 0$. We show that for each $0 \leq a \leq 2$ the Hausdorff dimension of $T_U(a)$ is almost surely $2-a$ and that $T_U(a)$ is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure considered by Duplantier and Sheffield. Joint work with Xiaoyu Hu and Yuval Peres