  We introduce the first passage set (FPS) of constant level $-a$ of the
two-dimensional continuum Gaussian free field (GFF) on finitely connected
domains. Informally, it is the set of points in the domain that can be
connected to the boundary by a path on which the GFF does not go below $-a$. It
is, thus, the two-dimensional analogue of the first hitting time of $-a$ by a
one-dimensional Brownian motion. We provide an axiomatic characterization of
the FPS, a continuum construction using level lines, and study its properties:
it is a fractal set of zero Lebesgue measure and Minkowski dimension 2 that is
coupled with the GFF $\Phi$ as a local set $A$ so that $\Phi+a$ restricted to
$A$ is a positive measure. One of the highlights of this paper is identifying
this measure as a Minkowski content measure in the non-integer gauge $r \mapsto
\vert\log(r)\vert^{1/2}r^{2}$, by using Gaussian multiplicative chaos theory.
