Recently, a European transport project has been carried out among several fusion devices for studying the possible link between the mean radial electric field (Er), long-range correlation (LRC) and edge bifurcations in fusion plasmas. The main results reported in this paper include: (i) the discovery of low-frequency LRCs in potential fluctuations which are amplified during the development of edge mean Er using electrode biasing and during the spontaneous development of edge sheared flows in stellarators and tokamaks. Evidence of nonlocal energy transfer and the geodesic acoustic mode modulation on local turbulent transport have also been observed. The observed LRCs are consistent with the theory of zonal flows described by a 'predator–prey' model. The results point to a significant link between the LRC and transport bifurcation. (ii) Comparative studies in tokamaks, stellarators and reversed field pinches have revealed significant differences in the level of the LRC. Whereas the LRCs are clearly observed in tokamaks and stellarators, no clear signature of LRCs was seen in the RFX-mod reversed field pinch experiments. These results suggest the possible influence of magnetic perturbations on the LRC, in agreement with recent observations in the resonant magnetic perturbation experiments at the TEXTOR tokamak. (iii) The degree of the LRCs is strongly reduced on approaching the plasma density-limit in tokamaks and stellarators, suggesting the possible role of collisionality or/and the impact of mean Er × B flow shear on zonal flows.