In this talk, aimed at the non-expert, I will describe a major breakthrough in probability theory: Based on pioneering work of Oded Schramm, it has been realized that a fairly large class of random curves associated with critical lattice processes from statistical physics can be described by a simple differential equation, the ``Schramm Loewner Evolution'' SLE, involving conformal maps. Examples include percolation, the loop erased random walk, and conjecturally the self-avoiding walk and the Ising model. As a consequence, numerous ``predictions'' from theoretical physics have been proved rigorously, several conjectures (e.g. Mandelbrots conjecture about the Hausdorff dimension of the ``Brownian frontier'') have been settled, and some outstanding problems (e.g. the size of self avoiding walks) moved within reach. I will also describe other families of random conformal maps, as well as open problems.