The nodal lines of random wavefunctions are investigated. We demonstrate numerically that they are well approximated by the so-called SLE6 curves which describe the continuum limit of the percolation cluster boundaries. This result gives additional support to the recent conjecture that the nodal domains of random (and chaotic) wavefunctions in the semi-classical limit are adequately described by the critical percolation theory. It is also shown that using the dipolar variant of SLE reduces significantly finite size effects.