One way of attempting to explain the prevalence of magnetic fields in natural large scale objects (planets, stars, galaxies) is the "kinematic dynamo problem": given an initially unmagnetized, flowing, electrically conducting fluid, will a small seed magnetic field tend to grow exponentially with time? If the answer is yes, then the fluid flow generates its own field. Recent work on this problem in the physically important regime of very high electrical conductivity has shown, somewhat surprisingly, that concepts of chaotic dynamics and fractals are the key to obtaining interesting solutions. This talk will explain some of these recent ideas.