The percolation properties of random fields arise naturally in many contexts, ranging from planet science to transport in disordered systems. In this talk we will discuss new results concerning the percolation transition of long-range correlated random fields. In particular we show that the level sets of surfaces with negative Hurst exponent are conformal fractals. Moreover, we revisit and solve the long-standing problem of the percolation transition in the 2D Gaussian Free Field (GFF). We show the existence of a non-trivial transition where the level set of the GFF become "logarithmic fractals". The correlation function of such fractals is also exactly computed.