The BK inequality says, roughly speaking, that the probability that two events `occur disjointly' is smaller than or equal to the product of the two individual probabilities. It has been used fruitfully in percolation and related topics. Until recently, no inequalities of this type were known for situations where the underlying random variables are dependent. However, in the last two years considerable progress has been made in this direction. This talk will start with an introduction and then highlight these new developments. Partly based on joint work with Johan Jonasson and joint work with Alberto Gandolfi.