Faraday waves are standing waves which arise through a parametric instability on the surface of a vertically oscillated fluid layer. They can emerge with various symmetries, simply square to $N$-fold rotationally symmetric, which for $N > 3$ are quasi-crystalline. In an experiment with a very large aspect ratio we determine the boundaries of the stability regions of waves with different rotational symmetries in the driving frequency–amplitude parameter plane. We find a remarkable agreement with a recent theory by Chen & Viñals (1999) who predict the stability boundaries at the onset amplitude. We argue why such agreement can only be observed in a very large experiment. The main nonlinear damping mechanism of the waves is a three-wave resonance. We devise a simple model that captures this mechanism and that can explain quantitatively the change of the symmetry of the waves with fluid depth. Detailed information about the surface is obtained by scanning the wave field and measuring the phase of subharmonic and harmonic components. Also the results of these measurements compare very favourably to the theoretical predictions.