Analysis on Fractals -- Heat Kernels on Metric Spaces with Doubling Measure -- Self-similarity and Random Walks -- Conformal Dynamics and Schramm-Loewner Evolution -- Multifractal Analysis of the Reverse Flow for the Schramm-Loewner Evolution -- Random Fractal Processes -- From Fractals and Probability to Lévy Processes and Stochastic PDEs -- Emergence of Fractals in Complex Systems -- A Survey of Dynamical Percolation -- Measure-valued Processes, Self-similarity and Flickering Random Measures -- Random Maps and Their Scaling Limits -- Iterated Function Schemes and Transformations of Fractals -- Transformations Between Fractals -- Geometric Realizations of Hyperbolic Unimodular Substitutions -- Random Cantor Sets and Their Projections.

Over the last fifteen years fractal geometry has established itself as a substantial mathematical theory in its own right. The interplay between fractal geometry, analysis and stochastics has highly influenced recent developments in mathematical modeling of complicated structures. This process has been forced by problems in these areas related to applications in statistical physics, biomathematics and finance. This book is a collection of survey articles covering many of the most recent developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals. The authors were the keynote speakers at the conference "Fractal Geometry and Stochastics IV" at Greifswald in September 2008.