Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of the underlying graph a random number (‘fitness’) is assigned and a path through the graph is called accessible if all random numbers along the path are in ascending order. After a brief introduction to the biological context, I will review recent results for the case of uncorrelated fitness values (the house-of-cards model) as well as for correlated models such as Kauffman's NK-model and the rough Mount Fuji model in which a constant fitness gradient is superimposed on an uncorrelated random background. The natural graph topology is that of an L-dimensional hypercube, but for some purposes it is useful to consider trees in which the height and the branching number are scaled to infinity. The talk is based on work with Jasper Franke, Stefan Nowak, and Benjamin Schmiegelt.