We consider level set percolation for the Gaussian free field on the Euclidean lattice in dimensions larger than or equal to three. It had previously been shown by Bricmont, Lebowitz, and Maes that the critical level is non-negative in any dimension and finite in dimension three. Rodriguez and Sznitman have extended this result by proving that it is finite in all dimensions, and positive in all large enough dimensions. We show that the critical parameter is positive in any dimension larger than or equal to three. In particular, this entails the percolation of sign clusters of the Gaussian free field. This talk is based on joint work with A. Prévost (Köln) and P.-F. Rodriguez (Los Angeles).