Percolation is a natural way of producing a random subgraph of a fixed graph: fix some p in (0,1), and then independently retain each edge with probability p and delete with probability 1-p. This is one of the simplest models exhibiting a “phase transition”; once p passes a certain threshold, the random subgraph suddenly switches from a “a very disconnected phase” to a “very connected phase.” I’ll start from the beginning and aim to prove two nice theorems which now have nice proofs: the uniqueness of the infinite cluster for amenable graphs (Burton-Keane) and the sharpness of the phase transition (the recent argument of Duminil-Copin and Tassion).