It has been predicted by physicists that the scaling limit of a number of models in statistical physics (percolation, self-avoiding random walk, uniform spanning trees, Ising model, loop-erased walks) in two dimensions "at criticality" are conformally invariant. This is a generalization of the fact that the scaling limit of simple random walk, i.e., Brownian motion, is conformally invariant in two dimensions. The conformal invariance assumption has allowed physicists to make nonrigorous exact predictions for critical exponents. In the last few years there has been a lot of progress in proving these conjectures. Many problems that seemed very difficult a few years ago have been solved. The most important new idea is the the Schramm-Loewner evolution (SLE) which is a random, conformally invariant curve. I will give an introduction to the Schramm-Loewner evolution and give a survey of the results on scaling limits. No previous knowledge of models from statistical physics will be assumed.