Ordinary (independent) percolation models have a sharp percolation transition: below the percolation threshold the cluster size distribution has exponential decay. For 2-dimensional models this was first proved by Kesten (1980). In 1981 Russo proved a so-called approximate zero-one law and pointed out that a key step in Kesten's argument can be seen as a special case of this more general law. A few years ago, new results by Bollobas and Riordan for the two-dimensional Voroinoi percolation model triggered more research in that direction. I will mainly focus on the contact process, a mathematical model of spatial epidemics, vegetation patterns and other natural random spatial structures.