Random growth processes are paradigmatic in probability theory and describe phenomena in many contexts, such as cancer growth, rumour propagation and population dynamics. In this talk, we will explore some classical growth processes and provide an overview for intriguing mathematical results related to them. In particular, we will see diffusion limited aggregation (DLA), a growth process that has defied a strong mathematical treatment for decades. We will discuss known results for DLA and its variations, culminating in an astonishing recent result by Elboim, Nam, and Sly that describes critical behaviour for the model in dimension one, the only dimension that is known to observe a phase transition.