There are many real-world processes exhibiting fractal growing shapes - from mineral deposition and coral growth to lightning strikes, and in many of them growth is related to diffusion properties. We will discuss two seminal models: Diffusion Limited Aggregation was introduced by Witten and Sanders in 1981 and was generalized to Dielectric Breakdown Model by Niemayer et al shortly afterwards. Numerically they approximate very well a wide range of physical phenomena. However, despite a very simple definition (DLA cluster grows by attaching particles undergoing Brownian motion when they hit the aggregate), very little is understood today, and even less is known rigorously - essentially, only the famous Harry Kesten upper bound on the DLA growth. We will try to show the flavor of these models and present some new results. Based on a joint preprint with Ilya Losev and some further work.