Conformal invariance, droplets, and entanglement

Very brief surveys are presented of three topics of importance for interacting random systems, namely conformal invariance, droplets, and entanglement. For case of description, the emphasis throughout is upon progress and open problems for the percolation model, rather than for the more general random-cluster model. Substantial recent progress has been made on each of these topics, as summarised here. Detailed bibliographies of recent work are included.