Fractal-like structures are common in nature. They have a property called self-similarity, which means that each potion looks like the whole. Some of the most famous mathematical fractals, such as the Cantor, Julia and Mandelbrot sets or the Sierpinski carpet, are produced by a deterministic process and contain identical, scaled-down copies of the themselves. On the contrary, natural fractal-like objects usually look random and are self-similar only in a statistical sense, which means that each portion of the object looks similar but not identical to the whole. This talk will explore how combining self-similarity and randomness produces extremely interesting objects, whose analysis requires a mixture of techniques from different areas of mathematics. Random fractals have many applications to various fields of mathematics, the natural sciences and economics. It will explain how they appear in such diverse contexts as the modeling of financial markets, the theory of phase transitions, and cosmology.