  We investigate the percolation phase transition for level sets of the
Gaussian free field on $\mathbb{Z}^d$, with $d\geqslant 3$, and prove that the
corresponding critical parameter $h_*(d)$ is strictly positive for all
$d\geqslant3$, thus settling an open question from arXiv:1202.5172. In
particular, this implies that the sign clusters of the Gaussian free field
percolate on $\mathbb{Z}^d$, for all $d\geqslant 3$. Among other things, our
construction of an infinite cluster above small, but positive level $h$
involves random interlacements at level $u>0$, a random subset of
$\mathbb{Z}^d$ with desirable percolative properties, introduced in
arXiv:0704.2560 in a rather different context, a certain Dynkin-type
isomorphism theorem relating random interlacements to the Gaussian free field,
see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two
objects, lifted to a continuous metric graph structure over $\mathbb{Z}^d$.
