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Abstract: The concept of self-organized criticality (SOC)has been proposed by physicists as a mechanism underlying the occurrence of various fractal structures in nature. The common feature of SOC models is that they possess a stochastic dynamics that drives them towards a `critical state' characterized by self-similarity. The talk will focus on two examples: the Abelian sandpile model and invasion percolation. Their common appeal is that they are simple to define, yet lead to difficult mathematical questions. Work of Majumdar and Dhar has revealed a connection of the Abelian sandpile to the uniform spanning tree. Using their insight and spanning tree techniques, I discuss the infinite volume limit of the model in dimensions two and higher, as well as the absence of infinite avalanches in dimensions d > 4. Invasion percolation is a stochastic growth process that reproduces the picture of critical Bernoulli percolation in a striking way. Its behaviour is closely related to Kesten's incipient infinite cluster. I will describe this relationship in the two-dimensional case.