One of the most fruitful melding of ideas in the last few decades has been in probability, complex analysis, mathematical physics, and combinatorics in making precise and rigorous the predictions from physics on (mostly) equilibrium models in statistical mechanics. These produce random fractals which can be viewed either as interesting objects in themselves or as aspects of a more complicated random geometry. I will give a survey of some of the main ideas with an emphasis on the idea of “exploration”, which is essentially the simple idea of predicting the nature of an object based on partial observation. I also hope to give an idea of current research in these areas. This talk will be intended for a general mathematics audience.