The temporal behaviour of turbulence associated with the L - H transition is described. In the ASDEX tokamak the level of density fluctuations in a narrow region inside the separatrix and in the scrape-off layer is quenched on a 100 timescale. After the L - H transition fluctuations of a different nature are destabilized which, however, show no marked influence on transport across the barrier. The time development of magnetic turbulence is closely correlated with density turbulence. A comparison is made with edge turbulence in the W7-AS stellarator H-mode where edge turbulence during the transition evolves on a similar timescale.