Diffusion Limited Aggregation: what we do and do not know about it

Diffusion Limited Aggregation is a very simple mathematical model which describes a wide range of natural phenomena. Despite its simplicity, there is very little progress in understanding its large-scale structure. Since its introduction by Witten and Sander over 40 years ago, there was only one mathematical result. In 1987 Kesten obtained an upper bound on the growth rate. In this talk I will discuss DLA and some related models and the recent progress in understanding DLA. In particular, a new simpler proof of Kesten result which generalizes to other aggregation models.