The simple random walk on the square grid in the plane converges to Brownian motion under appropriate scaling. Planar Brownian motion is rotationally invariant, while the random walk is not. In fact, Brownian motion enjoys conformal invariance, which is far richer. Many other random systems, such as critical percolation and the critical Ising model of magnetism, also seem to exhibit conformal invariance in the scaling limit. This conformal invariance implies that certain paths arising naturally from these processes fall into a particular one-parameter family of random fractal paths called Stochastic Loewner Evolution, or SLE.